201 Plane Trigonometry Terms and Definitions | Mathematics Board Exam Review

INTRODUCTION

Plane trigonometry is one of the most consistently tested subjects across all Philippine engineering licensure examinations. Whether you are reviewing for the ECE, EE, CE, ME, GeE, or any other PRC board exam, you will encounter problems that draw directly from the vocabulary and formulas of plane trigonometry. The subject is not just a standalone topic. It serves as the mathematical language of waves, circuits, vibrations, structural analysis, and surveying. If your trigonometry is weak, the effects ripple across multiple subjects on the exam.

The scope of plane trigonometry terms is broader than most reviewees initially realize. It begins with angle measurement and the definitions of the six trigonometric functions, but it extends into a rich network of identities, inverse functions, triangle solving methods, and wave behavior. Every one of these areas has been tested on past PRC board exams. The board examiners do not just ask you to compute a sine or cosine. They ask you to simplify complex expressions using identities, solve triangles with incomplete information, analyze periodic functions, and apply trigonometric principles to real engineering scenarios.

This glossary organizes 201 terms from plane trigonometry alphabetically so you can use it as a reference throughout your review. Each definition is written with the board exam in mind. We explain not just what a term means but how it functions in the context of solving problems under exam conditions. You will find classic terms like the Laws of Sines and Cosines alongside less obvious ones like the product to sum identities, the exsecant function, and the graphs of transformed trigonometric functions. All of them have appeared in Mathematics board exam materials at some point, and all of them are worth knowing.

Use this glossary actively, not passively. Read each definition, connect it to a formula or a problem type you have already practiced, and flag the terms you are least confident about for deeper review. The reviewees who perform best on the mathematics portion of the PRC board exams are those who understand the vocabulary of the subject deeply enough to recognize what a problem is asking before they even begin solving it. That level of recognition starts with knowing your terms cold.

The 201 Plane Trigonometry Terms and Definitions

1. 30-60-90 Triangle

A right triangle with interior angles of 30°, 60°, and 90°. The sides are in the ratio 1 : √3 : 2, where 1 is opposite the 30° angle, √3 is opposite the 60° angle, and 2 is the hypotenuse. This triangle is one of the two most important special triangles in trigonometry. All exact trigonometric values for 30° and 60° can be derived from this triangle without memorizing them separately.

2. 45-45-90 Triangle

A right triangle with interior angles of 45°, 45°, and 90°. The sides are in the ratio 1 : 1 : √2, where the two equal legs have length 1 and the hypotenuse has length √2. This is the isosceles right triangle. All exact trigonometric values for 45° are derived from this triangle. Along with the 30-60-90 triangle, it forms the foundation of exact value calculations in trigonometry and appears in countless board exam problems.

3. Absolute Value of a Trigonometric Expression

The non-negative value of a trigonometric expression, obtained by dropping any negative sign. When working with identities or equations involving absolute values of sine or cosine, remember that |sin θ| = sin θ when sin θ is non-negative and |sin θ| = −sin θ when sin θ is negative. This concept appears in problems involving amplitude, piecewise trigonometric functions, and the evaluation of expressions across multiple quadrants.

4. Acute Angle

An angle whose measure is greater than 0 degrees and less than 90 degrees. In a right triangle, the two non-right angles are always acute. Many trigonometric identities and special angle values used in board exam problems involve acute angles exclusively, so recognizing them quickly saves time during the exam.

5. Acute Triangle

A triangle in which all three interior angles are acute, meaning each angle measures less than 90 degrees. In an acute triangle, the square of the longest side is less than the sum of the squares of the other two sides. This relationship is useful when classifying triangles before applying the Law of Cosines.

6. Adjacent Side

In a right triangle, the side that is next to a given acute angle and is not the hypotenuse. The cosine of an angle equals the ratio of the adjacent side to the hypotenuse, and the tangent equals the ratio of the opposite side to the adjacent side. Keeping track of which side is adjacent versus opposite is essential for setting up trigonometric ratios correctly.

7. Ambiguous Case

A situation that arises in the Law of Sines when two sides and a non-included angle are given. Depending on the given measurements, the triangle may have no solution, exactly one solution, or two possible solutions. The ambiguous case is a classic board exam trap. Always check whether the given angle is acute or obtuse and compare the given side opposite the angle to the other given side before concluding how many triangles exist.

8. Amplitude

The maximum displacement of a trigonometric function from its midline or equilibrium position. For a function of the form y = A sin(Bx + C) + D, the amplitude is the absolute value of A. In board exam problems involving waves, oscillations, or AC circuits, the amplitude corresponds to the peak value of the signal. It is always a positive number.

9. Angle

The measure of rotation between two rays sharing a common endpoint called the vertex. Angles can be measured in degrees or radians. In trigonometry, angles can be positive (counterclockwise rotation) or negative (clockwise rotation), which matters when working with angles in standard position on the coordinate plane.

10. Angle Bisector

A ray that divides an angle into two equal parts. In a triangle, the angle bisector from a vertex divides the opposite side in the ratio of the two adjacent sides. This is the angle bisector theorem. Board exam problems involving angle bisectors often combine this ratio property with the Law of Sines or with trigonometric area formulas to find unknown side lengths or areas.

11. Angle of Depression

The angle measured downward from the horizontal line of sight to an object located below the observer. In board exam word problems, the angle of depression is almost always used in combination with right triangle trigonometry to find heights or horizontal distances. Note that the angle of depression from the observer equals the angle of elevation from the object, as they are alternate interior angles.

12. Angle of Elevation

The angle measured upward from the horizontal line of sight to an object located above the observer. Like the angle of depression, it appears frequently in board exam application problems involving towers, buildings, and hills. Always draw a diagram and identify the right triangle before setting up the trigonometric equation.

13. Angle of Inclination

The angle that a line makes with the positive x-axis, measured counterclockwise, ranging from 0 degrees to 180 degrees exclusive. The tangent of the angle of inclination equals the slope of the line. This concept connects trigonometry directly to analytic geometry and appears in problems involving lines and their orientations on the coordinate plane.

14. Angular Velocity

The rate at which an angle changes with respect to time, typically measured in radians per second. Angular velocity is denoted by the Greek letter omega. In problems involving circular motion or rotating machinery, angular velocity is related to linear velocity by the equation v = rω, where r is the radius of rotation. This term bridges trigonometry with physics and engineering mechanics.

15. Annex Angle

An older term sometimes seen in Mathematics engineering review materials that refers to the supplement of a given angle, or in some contexts, to the reference angle. When you encounter this term in legacy review books, identify the context carefully to determine whether it means 180 minus the angle or the acute reference angle in the appropriate quadrant.

16. Arc

A portion of the circumference of a circle. In trigonometry, arc length is calculated using the formula s = rθ, where r is the radius and θ is the central angle measured in radians. The arc is also foundational in the definition of radian measure itself. Board exam problems frequently ask for arc length given radius and central angle, or for the central angle given arc length and radius.

17. Arc Length Formula

The formula s = rθ, where s is the arc length, r is the radius of the circle, and θ is the central angle in radians. This formula is one of the most frequently used relationships in problems that combine trigonometry with circular motion or geometry. Always convert degrees to radians before applying this formula.

18. Arccosine

The inverse function of cosine, written as arccos(x) or cos⁻¹(x). It returns the angle whose cosine equals x. The range of arccosine is restricted to [0°, 180°] or [0, π] radians to ensure a unique output. In board exam problems, arccos is used to find missing angles when the cosine ratio is known.

19. Arcsecant

The inverse function of secant, written as arcsec(x) or sec⁻¹(x). It returns the angle whose secant equals x. The domain of arcsecant is all real numbers with absolute value greater than or equal to 1. Arcsecant appears in calculus-based board exam problems involving the integration and differentiation of inverse trigonometric functions.

20. Arcsine

The inverse function of sine, written as arcsin(x) or sin⁻¹(x). It returns the angle whose sine equals x. The range of arcsine is restricted to [−90°, 90°] or [−π/2, π/2] radians. When solving for angles in right triangle problems, arcsine is used when the ratio of the opposite side to the hypotenuse is known.

21. Arctangent

The inverse function of tangent, written as arctan(x) or tan⁻¹(x). It returns the angle whose tangent equals x. The range of arctangent is restricted to (−90°, 90°) or (−π/2, π/2) radians. Arctangent is commonly used in engineering problems involving slopes, phase angles, and direction angles.

22. Area of a Triangle (Trigonometric Form)

The area of a triangle expressed as (1/2)ab sin C, where a and b are two sides of the triangle and C is the included angle between them. This formula is used when the height of the triangle is not directly given. It is derived from the basic area formula and the definition of sine. Board examiners often give two sides and the included angle and ask for the area, making this formula essential.

23. Bearing

The direction of a line measured as an angle from a reference direction, typically north or south, toward east or west. In board exam problems, bearings appear in two formats: compass bearings (N 30° E) and azimuth bearings expressed as a single angle from 0° to 360°. Converting between these formats and applying trigonometry to navigation problems is a tested skill.

24. Central Angle

An angle formed at the center of a circle by two radii. The measure of the central angle equals the measure of the arc it intercepts. In trigonometry, the central angle in radians is used in the arc length formula and the sector area formula. Understanding the central angle is foundational for radian measure and circular trigonometry problems.

25. Circular Functions

Another name for the trigonometric functions when they are defined in terms of the unit circle rather than right triangle ratios. In this definition, for any angle θ in standard position, the terminal side intersects the unit circle at a point (cos θ, sin θ). Using the unit circle allows the trigonometric functions to be defined for all real angles, not just angles between 0° and 90°. The circular function definition is the more general and complete definition used in advanced mathematics.

26. Circumradius

The radius of the circle circumscribed about a triangle, denoted R. By the Law of Sines, the circumradius equals a/(2 sin A) = b/(2 sin B) = c/(2 sin C), where a, b, c are the sides and A, B, C are the opposite angles. The circumradius appears in board exam problems that ask for the radius of the circumscribed circle of a triangle, which is a common geometry-trigonometry hybrid problem.

27. Cofactor Expansion

In the context of trigonometry, cofactor expansion sometimes refers to expanding a determinant whose entries involve trigonometric functions. More broadly, the term reminds engineering reviewees that trigonometric expressions and matrices intersect in rotation matrices and transformation matrices, both of which appear in advanced engineering applications and in some board exam problems involving coordinate transformations.

28. Co-function Identities

A set of identities that relate each trigonometric function to a complementary function. The co-function identities state that sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ, and so on. The prefix “co” in cosine, cotangent, and cosecant literally stands for “complementary.” These identities are used frequently in simplifying expressions and verifying other identities during board exams.

29. Complementary Angles

Two angles whose measures add up to 90 degrees. In trigonometry, the sine of an angle equals the cosine of its complement, and the tangent of an angle equals the cotangent of its complement. Recognizing complementary angle pairs allows you to quickly convert between trigonometric functions when simplifying expressions in board exam problems.

30. Complementary Function

In the context of trigonometry and differential equations, the complementary function is the homogeneous solution of a differential equation. Its trigonometric relevance lies in the use of sine and cosine as solutions to second-order linear differential equations with constant coefficients. This connection is critical for electrical engineers dealing with RLC circuits and mechanical engineers dealing with vibrations.

31. Compound Angle

An angle that is expressed as the sum or difference of two simpler angles. The trigonometric functions of compound angles are evaluated using the sum and difference identities. For example, sin(75°) can be computed as sin(45° + 30°) using the sum identity. Recognizing a compound angle and applying the appropriate identity is a frequently tested skill on the engineering board exam.

32. Concyclic Points

Four or more points that all lie on the same circle. In trigonometric and geometric problems, showing that four points are concyclic often involves using the inscribed angle theorem or proving that opposite angles of the formed quadrilateral sum to 180 degrees. This concept appears in problems connecting plane geometry with trigonometric identities.

33. Conditional Trigonometric Equation

A trigonometric equation that is true only for specific values of the variable, not for all values. This distinguishes it from a trigonometric identity, which is true for all allowable values. Solving a conditional equation means finding the specific angles that satisfy it within a given interval. Recognizing the difference between an identity and a conditional equation is an important conceptual distinction tested on the board exam.

34. Congruent Triangles

Triangles that have exactly the same size and shape, meaning all corresponding sides and angles are equal. The conditions for triangle congruence:  SSS, SAS, ASA, AAS, and HL are also determine when a triangle is uniquely solvable using trigonometric methods. In board exam problems, proving congruence is the geometric foundation for establishing that two trigonometric expressions are equal.

35. Cosecant

The reciprocal of the sine function, abbreviated as csc. For an angle θ, csc θ = 1/sin θ = hypotenuse/opposite side. The cosecant function is undefined when sin θ = 0, which occurs at multiples of 180 degrees. In board exam problems, cosecant appears in trigonometric identities and in problems involving reciprocal functions.

36. Cosecant and Secant as Phase Shifted Functions

The relationship that expresses cosecant and secant as phase shifted versions of sine and cosine, since csc θ = sec(90° − θ) and sec θ = csc(90° − θ). This follows directly from the co function identities applied to reciprocal functions. Recognizing this relationship helps when simplifying expressions that mix cosecant and secant with sine and cosine, and reinforces the complementary angle pattern that appears throughout the reciprocal and co function identities tested on the board exam.

37. Cosecant Graph

The graph of the cosecant function, which is the reciprocal of the sine graph. The cosecant graph has vertical asymptotes wherever sine equals zero, meaning at every integer multiple of π. Between the asymptotes, the graph forms U-shaped curves opening upward (when sine is positive) or downward (when sine is negative). The minimum value of csc θ where it is positive is 1, and the maximum where it is negative is −1. Board exam problems that involve sketching or analyzing reciprocal function graphs require understanding this structure.

38. Cosine

One of the six fundamental trigonometric functions. In a right triangle, cosine of an angle equals the ratio of the adjacent side to the hypotenuse. On the unit circle, cosine corresponds to the x-coordinate of the point on the circle. The cosine function has a range of [−1, 1] and a period of 360 degrees or 2π radians. It is one of the most frequently used functions in all of engineering mathematics.

39. Cosine Rule

See Law of Cosines. The cosine rule is the more commonly used informal name for this law in Mathematics engineering review materials and textbooks.

40. Cotangent

The reciprocal of the tangent function, abbreviated as cot. For an angle θ, cot θ = cos θ/sin θ = adjacent side/opposite side. The cotangent is undefined when sin θ = 0, which occurs at integer multiples of 180 degrees. Cotangent appears frequently in trigonometric identities and in calculus-based engineering problems.

41. Cotangent Graph

The graph of the cotangent function, which has vertical asymptotes at every integer multiple of π (where sine equals zero) and zeros at odd multiples of π/2. The cotangent graph is a reflection and phase shift of the tangent graph. The period of cotangent is π, the same as tangent. Understanding the cotangent graph is less commonly tested than sine, cosine, and tangent but appears in comprehensive trigonometry sections of advanced board exam reviews.

42. Coterminal Angles

Two or more angles that share the same initial side and the same terminal side when drawn in standard position. Coterminal angles differ by multiples of 360 degrees or 2π radians. For example, 30° and 390° are coterminal. When a given angle is outside the standard range, finding its coterminal angle within [0°, 360°] helps you identify the correct reference angle and quadrant quickly.

43. Coversine

A less common trigonometric function defined as coversine(θ) = 1 − sin θ. Also written as covers(θ). This function appears in older Mathematics engineering board exam review books and in some navigation problems. While rarely seen in modern materials, it is worth recognizing if you encounter it in legacy review resources or historical exam items.

44. Degree

A unit of angle measurement equal to 1/360 of a full rotation. One degree is subdivided into 60 minutes, and one minute is subdivided into 60 seconds. The degree is the most commonly used unit of angle measurement in Mathematics engineering board exam problems, though radians are required in calculus-based problems and formula applications involving arc length and angular velocity.

45. Degree-Minute-Second Format

The format for expressing angles as a whole number of degrees plus a whole number of minutes plus a decimal or whole number of seconds. Written as D°M’S”. This format is standard in surveying, navigation, and precise geodetic work. Converting from decimal degrees to DMS and back is a regularly tested calculation skill in the Mathematics engineering board exam, especially for civil and geodetic engineering candidates.

46. Depression Angle

See Angle of Depression. This alternative term appears in some Mathematics review books and engineering references.

47. Difference Identities

Trigonometric identities that express the sine, cosine, or tangent of the difference of two angles in terms of functions of the individual angles. The key formulas are: sin(A − B) = sin A cos B − cos A sin B, cos(A − B) = cos A cos B + sin A sin B, and tan(A − B) = (tan A − tan B)/(1 + tan A tan B). These identities are tested directly in board exams and are also used to derive other identities.

48. Diagonal of a Polygon (Trigonometric)

The line segment connecting two non-adjacent vertices of a polygon. In trigonometric problems, the diagonals of regular polygons are computed using combinations of the Law of Cosines and the properties of isosceles triangles. For a regular polygon with n sides of length a, the diagonal formulas involve the sine of multiples of (180/n) degrees. These problems appear in the geometry-trigonometry crossover section of board exams.

49. Direction Angle

The angle a vector makes with the positive x-axis, measured counterclockwise. Computing the direction angle of a vector (a, b) requires evaluating arctan(b/a) and then adjusting based on the quadrant of the vector. Direction angles are used in problems involving force vectors, displacement, velocity, and other physical quantities represented as vectors. The connection between direction angle and the tangent function is direct and is a tested topic in engineering mathematics.

50. Domain of Inverse Trigonometric Functions

The set of input values for which an inverse trigonometric function is defined. The domain of arcsin and arccos is [−1, 1]. The domain of arctan and arccot is all real numbers. The domain of arcsec and arccsc is the set of real numbers with absolute value greater than or equal to 1. Knowing these domains is necessary for determining whether a given inverse trigonometric expression is defined and for setting up valid equations in board exam problems.

51. Double Angle Applications

Practical uses of the double angle identities in engineering problems. These include finding exact values of trigonometric functions at twice a given angle, simplifying expressions in electrical circuit analysis, solving quadratic-type trigonometric equations, and computing power in AC circuits using the identity P = (V²/R) × (1 + cos 2ωt)/2. Being able to connect the algebraic identity to its physical application is a higher-order skill tested in advanced board exam items.

52. Double Angle Identities

Identities that express trigonometric functions of twice an angle in terms of functions of the original angle. The most important are: sin(2θ) = 2 sin θ cos θ, cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ, and tan(2θ) = 2 tan θ/(1 − tan²θ). These identities appear constantly in board exam problems involving simplification, equation solving, and integration.

53. Elevation Angle

See Angle of Elevation.

54. Equal Angles in a Triangle

A property stating that if two sides of a triangle are equal, the angles opposite those sides are also equal, and vice versa. This is the isosceles triangle theorem. In trigonometric terms, if a = b in a triangle, then sin A = sin B, which means A = B or A + B = 180°. Since A and B are interior angles of a triangle, only A = B is valid. This reasoning applies when solving triangles using the Law of Sines and checking solutions.

55. Equation of a Circle (Trigonometric Parametric Form)

The parametric equations x = r cos θ and y = r sin θ, which describe all points on a circle of radius r centered at the origin as θ varies from 0° to 360°. This is the trigonometric parametric representation of a circle and is used in calculus, engineering graphics, and problems involving motion along a circular path. The parameter θ directly corresponds to the angle in standard position, connecting parametric curves to circular trigonometry.

56. Equilateral Triangle

A triangle in which all three sides are equal in length and all three angles measure exactly 60 degrees. Trigonometric problems involving equilateral triangles often use the fact that splitting one in half produces two 30-60-90 right triangles. Recognizing this allows you to solve for side lengths and heights quickly without using the Law of Cosines.

57. Even Function

A function f(x) for which f(−x) = f(x) for all x in its domain. Among the six trigonometric functions, cosine and secant are even functions. Knowing this property means that cos(−θ) = cos θ, which is useful in simplifying expressions involving negative angles in board exam problems.

58. Exact Values

The trigonometric function values that can be expressed as simple fractions or radicals without a calculator. Exact values exist for the special angles 0°, 30°, 45°, 60°, and 90°, and for all angles coterminal with or supplementary to these. In Mathematics engineering board exams, problems are often designed so that a candidate who knows the exact values can solve them more quickly and accurately than one relying on decimal approximations. Knowing exact values fluently is a competitive advantage on timed exams.

59. Excircle

A circle that is tangent to one side of a triangle and to the extensions of the other two sides. A triangle has three excircles. The radius of an excircle (exradius) is denoted r₁, r₂, or r₃. The exradius formula r₁ = Area/(s − a), where s is the semi-perimeter, is the most commonly tested form on the Mathematics engineering board exam.

60. Exponent Rules Applied to Trigonometric Functions

The careful application of algebraic exponent rules when working with expressions like sin²θ or cos³θ. A common board exam trap is treating (sin θ)² as sin(θ²), which is incorrect. The expression sin²θ means (sin θ)², not sin(θ²). Similarly, sin⁻¹θ means arcsin θ, not 1/sin θ. Correctly interpreting trigonometric notation with exponents prevents errors in both identity manipulation and equation solving.

61. Exsecant

A less common trigonometric function defined as exsecant(θ) = sec θ − 1. It appears in older engineering references, particularly in highway curve problems and some navigation applications. Like coversine, it is not commonly tested in modern board exams but may appear in legacy review materials.

62. Extended Law of Sines

The version of the Law of Sines that explicitly includes the circumradius: a/sin A = b/sin B = c/sin C = 2R, where R is the radius of the circumscribed circle. This extended form is used in problems where the circumradius is either given or required. The extended Law of Sines connects triangle solving with circle geometry and appears in combined geometry-trigonometry problems on the engineering board exam.

63. Five-Part Formula

An alternative name sometimes used in older Mathematics references for the Law of Tangents or for a set of spherical trigonometry formulas. When you encounter this term in review materials, check the context carefully to determine which specific formula is being referenced. In plane trigonometry, the most likely interpretation involves the relationship between the tangents of half-angle sums and differences and the ratio of the corresponding side sums and differences.

64. Fixed Point

In trigonometric graphs, a fixed point is a value of x for which a transformed version of a trigonometric function has the same output as the original untransformed function. Fixed point analysis helps in understanding the relationship between parent and transformed trigonometric functions. This concept occasionally appears in advanced board exam problems involving function transformations.

65. Frequency

The number of complete cycles of a periodic function per unit of time or per unit of the independent variable. For a function of the form y = A sin(Bx), the frequency is B/(2π) cycles per unit. In engineering problems involving alternating current, vibrations, and waves, frequency is one of the primary parameters describing the behavior of the system.

66. Frequency (Angular)

Also called angular frequency, this is the rate of oscillation in radians per unit time, denoted by ω (omega). It is related to the ordinary frequency f by the equation ω = 2πf. In electrical engineering, angular frequency is the coefficient of t in expressions like V(t) = V₀ sin(ωt + φ). Board exam problems involving AC circuits consistently use angular frequency, and converting between f and ω is a routine calculation in these problems.

67. Fundamental Identities

The set of basic trigonometric identities from which all other identities can be derived. These include the reciprocal identities (csc = 1/sin, sec = 1/cos, cot = 1/tan), the quotient identities (tan = sin/cos, cot = cos/sin), and the Pythagorean identities (sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ). Mastering these fundamental identities is the foundation of success in all trigonometry board exam problems.

68. General Solution of a Trigonometric Equation

The complete set of all solutions of a trigonometric equation, expressed using the periodicity of the function. For sin θ = k, the general solution is θ = arcsin(k) + 360°n or θ = 180° − arcsin(k) + 360°n, where n is any integer. For cos θ = k, the general solution is θ = ±arccos(k) + 360°n. The general solution is more comprehensive than the principal value solution and is expected in board exam problems that ask for all solutions.

69. Geometric Mean (Trigonometric Context)

In a right triangle, the altitude from the right angle to the hypotenuse is the geometric mean between the two segments it creates on the hypotenuse. This relationship is expressed trigonometrically as h = √(p × q), where p and q are the two hypotenuse segments. This property of right triangles is a tested concept in combined geometry-trigonometry problems on the board exam.

70. Grade (Angle Measurement)

A unit of angle measurement where a full circle is divided into 400 grades, also called gradians or gons. One grade equals 0.9 degrees and one right angle equals 100 grades. While degrees and radians are far more common in Mathematics board exams, grades appear occasionally in older problems and international references. Knowing how to convert between grades, degrees, and radians is a useful skill.

71. Graph of Cosine

The visual representation of the cosine function plotted on a coordinate system. The cosine graph starts at its maximum value of 1 when x = 0, descends to 0 at x = π/2, reaches its minimum of −1 at x = π, returns to 0 at x = 3π/2, and completes one full cycle at x = 2π. The cosine curve is a horizontal shift of the sine curve by π/2 units to the left. Board exam problems on graphing ask you to identify amplitude, period, phase shift, and vertical shift from equations or graphs.

72. Graph of Sine

The visual representation of the sine function plotted on a coordinate system. The sine graph starts at 0, rises to 1 at π/2, returns to 0 at π, drops to −1 at 3π/2, and completes one cycle at 2π. The sine function is odd and symmetric about the origin. Understanding the basic shape of the sine graph is essential for analyzing transformed functions in board exam problems.

73. Graph of Tangent

The visual representation of the tangent function. The tangent graph has vertical asymptotes at odd multiples of π/2, passes through the origin, and has a period of π (not 2π like sine and cosine). The tangent curve increases without bound as x approaches an asymptote from the left and decreases from positive infinity as x moves past the asymptote from the right. This graph behavior appears in problems involving phase angles and frequency response in engineering.

74. Half Angle Identities

Identities that express trigonometric functions of half an angle in terms of functions of the original angle. The key formulas are: sin(θ/2) = ±√[(1 − cos θ)/2], cos(θ/2) = ±√[(1 + cos θ)/2], and tan(θ/2) = (1 − cos θ)/sin θ = sin θ/(1 + cos θ). The sign depends on the quadrant of θ/2. These identities are tested in board exams and are essential in integral calculus problems involving trigonometric substitution.

75. Half-Perimeter

Another term for the semi-perimeter of a triangle, equal to s = (a + b + c)/2. Used in Heron’s formula and in the formulas for the inradius and exradii. Some Mathematics review books use “half-perimeter” while others use “semi-perimeter.” Both terms refer to the same quantity, and both appear in formulas you will need on the board exam.

76. Haversine

A trigonometric function defined as haversine(θ) = (1 − cos θ)/2 = sin²(θ/2). The haversine formula is used in navigation to calculate great circle distances between two points on a sphere given their latitudes and longitudes. While not commonly tested by name in modern Mathematics board exams, understanding its connection to the half angle identity for cosine is conceptually valuable.

77. Height of a Triangle (Trigonometric)

The perpendicular distance from a vertex to the opposite side (or its extension), also called the altitude. For a triangle with sides a, b, and included angle C, the altitude from the vertex at C to side c is h = a sin B = b sin A. Computing the height using the sine function is a direct application of right triangle trigonometry within an oblique triangle context and appears frequently in area-related board exam problems.

78. Heron’s Formula

A formula for computing the area of a triangle when all three side lengths are known: Area = √[s(s − a)(s − b)(s − c)], where s is the semi-perimeter and a, b, c are the side lengths. Heron’s formula is the standard method used in board exam problems when no angle is given and only the three sides are provided. It is derived from the area formula (1/2)ab sin C using the Law of Cosines to express sin C in terms of the sides.

79. Horizontal Shift

See Phase Shift. In trigonometry, horizontal shift and phase shift refer to the same displacement of the graph along the x-axis. Some Mathematics textbooks use “horizontal shift” to describe the displacement of the graph and reserve “phase shift” specifically for the physical interpretation in wave and signal problems.

80. Hypotenuse

The longest side of a right triangle, located opposite the right angle. In a right triangle with legs a and b and hypotenuse c, the Pythagorean theorem states c² = a² + b². The hypotenuse is the denominator in the definitions of both sine (opposite over hypotenuse) and cosine (adjacent over hypotenuse). Correctly identifying the hypotenuse is the first step in setting up any right triangle trigonometry problem.

81. Identity Verification

The process of proving that a given trigonometric equation is an identity, meaning it is true for all allowable values of the variable. Board exam problems that ask you to verify an identity require you to transform one side of the equation using known identities until it matches the other side. The standard approach is to work on the more complex side and simplify. Never add the same expression to both sides or cross-multiply when verifying identities.

82. Included Angle

The angle formed between two known sides of a triangle. When two sides and their included angle are given (SAS configuration), the Law of Cosines is the correct tool for solving the triangle. The included angle is specifically distinguished from a non-included angle because the two cases require entirely different solution methods, and confusing them is a common source of errors on the board exam.

83. Initial Side

The fixed ray from which an angle is measured in standard position. The initial side of an angle in standard position always lies along the positive x-axis. The angle is measured from the initial side to the terminal side, with counterclockwise rotation being positive and clockwise rotation being negative.

84. Initial Value of a Sinusoidal Function

The value of a sinusoidal function at x = 0 or t = 0. For y = A sin(C) + D, the initial value is A sin(C) + D. For y = A cos(C) + D, the initial value is A cos(C) + D. In engineering problems involving AC circuits or mechanical oscillations, the initial value represents the starting condition of the system and determines the phase angle required to match a given physical scenario.

85. Inradius

The radius of the circle inscribed within a triangle, denoted r. The inradius is given by r = Area/s, where s is the semi-perimeter of the triangle. It is also expressed as r = (s − a) tan(A/2) = (s − b) tan(B/2) = (s − c) tan(C/2). The inradius appears in board exam problems asking for the radius of the inscribed circle, which is a common combined trigonometry-geometry problem type.

86. Inscribed Angle

An angle formed by two chords of a circle that share an endpoint on the circle. The inscribed angle theorem states that an inscribed angle is equal to half the central angle that subtends the same arc. In board exam problems combining circle geometry with trigonometry, the inscribed angle theorem is frequently used to find unknown angles or to prove relationships between angles and arcs.

87. Intercept Form (Trigonometric Application)

The use of trigonometric principles to find where a line or curve crosses the axes. In right triangle and oblique triangle problems involving lines at angles, the intercepts with horizontal or vertical reference lines can be computed using the tangent or sine of the angle of inclination. This application appears in surveying and structural engineering problems where the intercept of a sloped surface or beam is required.

88. Interior Angle

An angle formed inside a polygon by two adjacent sides. In a triangle, the three interior angles always sum to 180 degrees. In trigonometry, the interior angles of a triangle are the A, B, and C used in the Law of Sines and Law of Cosines. Correctly identifying all three interior angles and their opposite sides is the essential setup step before applying any triangle-solving formula.

89. Inverse Cosine

See Arccosine. Also written as cos⁻¹. The inverse cosine function returns the angle in [0°, 180°] whose cosine equals the given value. In board exam problems, inverse cosine is used to find an angle when the adjacent side and hypotenuse are known, or when the cosine ratio is determined by the Law of Cosines.

90. Inverse Sine

See Arcsine. Also written as sin⁻¹. The inverse sine function returns the angle in [−90°, 90°] whose sine equals the given value. In board exam problems, inverse sine is used to find an angle when the opposite side and hypotenuse are known in a right triangle, or when the sine ratio is determined by other means.

91. Inverse Tangent

See Arctangent. Also written as tan⁻¹. The inverse tangent function returns the angle in (−90°, 90°) whose tangent equals the given value. It is widely used in engineering for computing angles from slope ratios, direction angles from vector components, and phase angles in AC circuit analysis.

92. Inverse Trigonometric Functions

The set of functions that return the angle corresponding to a given trigonometric ratio. The six inverse functions are arcsin, arccos, arctan, arccsc, arcsec, and arccot. Each has a restricted range to ensure a unique output. In board exam problems, inverse functions are used to solve for missing angles in triangles and in equations. They also appear in integration formulas in calculus-based engineering problems.

93. Isosceles Right Triangle

A right triangle with two equal legs and a 90° angle, which is the 45-45-90 triangle. The two equal legs are each of length 1 when the hypotenuse is √2. This triangle produces the exact trigonometric values for 45°. It appears frequently in board exam problems because its equal-leg symmetry often simplifies calculations significantly.

94. Isosceles Triangle

A triangle in which two sides are equal in length. The angles opposite the equal sides are also equal. Trigonometric problems involving isosceles triangles often exploit symmetry to reduce the number of unknowns. The altitude from the apex of an isosceles triangle bisects the base and creates two congruent right triangles, which allows direct application of right triangle trigonometry.

95. Law of Cosines

A generalization of the Pythagorean theorem for any triangle. The formula is: c² = a² + b² − 2ab cos C, where a, b, c are the side lengths and C is the angle opposite side c. The Law of Cosines is used when two sides and the included angle are given (SAS) or when all three sides are given (SSS). It reduces to the Pythagorean theorem when C = 90°, which is a useful check.

96. Law of Sines

The relationship between the sides and angles of any triangle expressed as a/sin A = b/sin B = c/sin C. The Law of Sines is used when two angles and one side are given (AAS or ASA) or when two sides and a non-included angle are given (SSA, the ambiguous case). Each ratio in the Law of Sines also equals the diameter of the circumscribed circle (2R), a fact occasionally tested on the board exam.

97. Law of Tangents

A formula that states: (a − b)/(a + b) = tan[(A − B)/2] / tan[(A + B)/2]. This law is an alternative to the Law of Cosines when two sides and the included angle are given, and it can be computationally more efficient in some cases. While it appears less often than the Laws of Sines and Cosines on modern board exams, it is worth knowing as a verification tool and for completeness in advanced trigonometry review. The Law of Tangents occasionally appears in advanced problems and in some older Mathematics board exam items.

98. Linear Combination of Sine and Cosine

An expression of the form a sin θ + b cos θ, which can always be rewritten as R sin(θ + φ) or R cos(θ − ψ), where R = √(a² + b²) and φ = arctan(b/a). This transformation is used in board exam problems that require finding the maximum or minimum value of such an expression, which equals R and −R respectively. It also appears in AC circuit analysis when combining voltage phasors.

99. Locus

The set of all points satisfying a given geometric condition. In trigonometric problems, loci involving circles, angles, and distances are expressed using trigonometric equations. For example, the locus of points at a fixed distance from a center is a circle, described parametrically as (r cos θ, r sin θ). Locus problems that combine geometry and trigonometry appear in comprehensive engineering mathematics reviews.

100. Maximum and Minimum Values

For the function y = A sin(Bx + C) + D, the maximum value is |A| + D and the minimum value is −|A| + D. These occur when the sine (or cosine) function equals 1 or −1 respectively. Identifying the maximum and minimum values of a trigonometric function is a direct board exam skill, and it connects to engineering problems where you need to find peak voltages, maximum stresses, or extreme displacements of a periodic system.

201. Zero of a Trigonometric Function

A value of θ for which a trigonometric function equals zero. Sine is zero at 0°, 180°, and their multiples. Cosine is zero at 90°, 270°, and their multiples. Tangent is zero wherever sine is zero (and cosine is non-zero). Knowing the zeros of each function helps solve trigonometric equations quickly and is essential for sketching accurate graphs of trigonometric functions.

CONCLUSION

Plane trigonometry is a subject with genuine depth, and the 201 terms in this glossary reflect that depth fully. For the board exam, your highest priority terms are the ones that appear in nearly every examination: the six trigonometric functions and their reciprocal relationships, the three Pythagorean identities, the sum and difference identities, the double angle identities, and the Laws of Sines and Cosines. These are the tools that unlock the majority of trigonometry problems in the PRC examinations, regardless of which engineering program you are in. If you can apply these fluently and correctly, you are already in a strong position for the mathematics portion of the board exam.

Pay special attention to the problem types that have historically caused the most errors among examinees. The ambiguous case of the Law of Sines has appeared repeatedly in past board exams and consistently traps candidates who rush into a calculation without checking the number of valid solutions. The phase shift and period of transformed sinusoidal functions are frequently tested in problems that look like simple graphing exercises but require precise parameter identification. Inverse trigonometric functions, particularly the restriction of their ranges, trip up examinees who confuse the principal value with the general solution. These are the areas where disciplined preparation pays the highest return.

Finally, do not treat plane trigonometry as a topic you finish and then set aside. It is the mathematical backbone of multiple engineering subjects, including AC circuit analysis, structural mechanics, vibration analysis, geodetic surveying, and signal processing. Every time you encounter a problem involving angles, waves, or rotation in any other subject area, you are applying plane trigonometry. The reviewee who deeply understands the vocabulary and tools of this subject carries a permanent advantage across the entire board exam, not just in the pure mathematics section. Use this glossary as a living reference throughout your entire review cycle, not just during your trigonometry week.

For practice problems on all these topics, head over to our Plane Trigonometry Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step by step solutions, organized by topic so you can drill exactly what you need to work on.

Keep reviewing. You are closer than you think.