Learn & Review: Trigonometry Final Exam

Jan 23, 2026

Trigonometry Final Exam Review

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Trigonometry Problem Solving Guide

This summary outlines key concepts and problem-solving techniques for basic trigonometry, covering angle conversions, coterminal angles, arc length, trigonometric ratios, identities, and reference angles.

1. Angle Conversions: Degrees to Radians and Vice Versa

  • Degrees to Radians: Multiply the degree measure by $\frac{\pi}{180^\circ}$.
    • Example: To convert $60^\circ$ to radians: $60^\circ \times \frac{\pi}{180^\circ} = \frac{60\pi}{180} = \frac{\pi}{3}$ radians.
  • Radians to Degrees: Multiply the radian measure by $\frac{180^\circ}{\pi}$.
    • Example: To convert $-\frac{5\pi}{6}$ to degrees: $-\frac{5\pi}{6} \times \frac{180^\circ}{\pi} = -\frac{5 \times 180}{6} = -5 \times 30 = -150^\circ$.

2. Coterminal Angles

Coterminal angles share the same terminal side when drawn in standard position. They can be found by adding or subtracting multiples of $360^\circ$ (or $2\pi$ radians).

  • To find a coterminal angle: Add or subtract $2\pi$ radians.
    • Example: For $\frac{5\pi}{8}$:
      • Adding $2\pi$: $\frac{5\pi}{8} + 2\pi = \frac{5\pi}{8} + \frac{16\pi}{8} = \frac{21\pi}{8}$.
      • Subtracting $2\pi$: $\frac{5\pi}{8} - 2\pi = \frac{5\pi}{8} - \frac{16\pi}{8} = -\frac{11\pi}{8}$.

3. Arc Length

The length of an arc ($s$) is calculated using the formula: $s = \theta \times r$, where $\theta$ is the central angle in radians and $r$ is the radius.

  • Steps:
    1. Convert the angle from degrees to radians if necessary.
    2. Multiply the angle in radians by the radius.
  • Example: For an angle of $120^\circ$ and a radius of 9 inches:
    1. Convert $120^\circ$ to radians: $120^\circ \times \frac{\pi}{180^\circ} = \frac{12\pi}{18} = \frac{2\pi}{3}$ radians.
    2. Calculate arc length: $s = \frac{2\pi}{3} \times 9 = 6\pi$ inches.

4. Right Triangle Trigonometry (SOH CAH TOA)

This mnemonic helps remember the basic trigonometric ratios:

  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

Key Concepts:

  • Hypotenuse: The side opposite the right angle (always the longest side).

  • Opposite: The side across from the angle in question.

  • Adjacent: The side next to the angle in question (not the hypotenuse).

  • Pythagorean Theorem: $a^2 + b^2 = c^2$ (where $a$ and $b$ are legs and $c$ is the hypotenuse) is used to find missing side lengths.

  • Special Right Triangles: Familiarity with common Pythagorean triples (e.g., 3-4-5, 5-12-13, 7-24-25, 8-15-17) can speed up calculations.

  • Example: For a right triangle with sides 3, 4, and hypotenuse 5, and angle $x$:

    • If $x$ is opposite the side of length 4:
      • $\sin x = \frac{4}{5}$
      • $\cos x = \frac{3}{5}$
      • $\tan x = \frac{4}{3}$

5. Reciprocal Identities

These identities relate the basic trigonometric functions:

  • Cosecant ($\csc \theta$) = $\frac{1}{\sin \theta}$

  • Secant ($\sec \theta$) = $\frac{1}{\cos \theta}$

  • Cotangent ($\cot \theta$) = $\frac{1}{\tan \theta}$

  • Tangent ($\tan \theta$) = $\frac{\sin \theta}{\cos \theta}$

  • Example: To find $\sec x$ when $\cos x = \frac{12}{13}$:

    • $\sec x = \frac{1}{\cos x} = \frac{1}{12/13} = \frac{13}{12}$.

6. Trigonometric Functions in Different Quadrants

The signs of trigonometric functions depend on the quadrant in which the angle lies:

  • Quadrant I (0° to 90°): All functions (sin, cos, tan) are positive.
  • Quadrant II (90° to 180°): Sine is positive; Cosine and Tangent are negative.
  • Quadrant III (180° to 270°): Tangent is positive; Sine and Cosine are negative.
  • Quadrant IV (270° to 360°): Cosine is positive; Sine and Tangent are negative.

Mnemonic: "All Students Take Calculus" (ASTC) helps remember which function is positive in each quadrant.

  • Example: If $\tan x = -\frac{8}{15}$ and $x$ is between $\frac{3\pi}{2}$ and $2\pi$ (Quadrant IV), then $\sin x$ is negative and $\cos x$ is positive. Using the 8-15-17 triangle, $\sin x = -\frac{8}{17}$ and $\csc x = -\frac{17}{8}$.

7. Pythagorean Identities

These fundamental identities relate squares of trigonometric functions:

  • $\sin^2 \theta + \cos^2 \theta = 1$

  • $1 + \tan^2 \theta = \sec^2 \theta$

  • $1 + \cot^2 \theta = \csc^2 \theta$

  • Example: If $\sin x = \frac{5}{7}$ and $x$ is in Quadrant II:

    1. Use $\sin^2 x + \cos^2 x = 1$.
    2. $(\frac{5}{7})^2 + \cos^2 x = 1 \implies \frac{25}{49} + \cos^2 x = 1$.
    3. $\cos^2 x = 1 - \frac{25}{49} = \frac{24}{49}$.
    4. $\cos x = \pm \sqrt{\frac{24}{49}} = \pm \frac{\sqrt{24}}{7} = \pm \frac{2\sqrt{6}}{7}$.
    5. Since $x$ is in Quadrant II, $\cos x$ is negative. So, $\cos x = -\frac{2\sqrt{6}}{7}$.

8. Even and Odd Trigonometric Functions

These properties describe how trigonometric functions behave with negative inputs:

  • Odd Functions: $\sin(-\theta) = -\sin \theta$, $\csc(-\theta) = -\csc \theta$, $\tan(-\theta) = -\tan \theta$, $\cot(-\theta) = -\cot \theta$.

  • Even Functions: $\cos(-\theta) = \cos \theta$, $\sec(-\theta) = \sec \theta$.

  • Example: $\sin(-\frac{\pi}{3}) = -\sin(\frac{\pi}{3})$.

9. Cofunction Identities

These identities relate a trigonometric function of an angle to the cofunction of its complement. They are often expressed using $90^\circ$ or $\frac{\pi}{2}$ radians.

  • $\sin \theta = \cos(90^\circ - \theta)$ or $\sin \theta = \cos(\frac{\pi}{2} - \theta)$

  • $\cos \theta = \sin(90^\circ - \theta)$ or $\cos \theta = \sin(\frac{\pi}{2} - \theta)$

  • $\tan \theta = \cot(90^\circ - \theta)$ or $\tan \theta = \cot(\frac{\pi}{2} - \theta)$

  • And so on for cotangent, secant, and cosecant.

  • Example: $\sin(\frac{\pi}{5}) = \cos(\frac{\pi}{2} - \frac{\pi}{5}) = \cos(\frac{5\pi}{10} - \frac{2\pi}{10}) = \cos(\frac{3\pi}{10})$.

10. Terminal Side of an Angle and Points on the Terminal Side

  • The terminal side is the ray where an angle measurement ends.

  • If a point $(x, y)$ lies on the terminal side of an angle $x$, the trigonometric functions can be found using $r = \sqrt{x^2 + y^2}$ (the distance from the origin, always positive):

    • $\sin x = \frac{y}{r}$
    • $\cos x = \frac{x}{r}$
    • $\tan x = \frac{y}{x}$
    • $\cot x = \frac{x}{y}$
    • $\sec x = \frac{r}{x}$
    • $\csc x = \frac{r}{y}$

  • Example: For point $(-40, -9)$:

    • $r = \sqrt{(-40)^2 + (-9)^2} = \sqrt{1600 + 81} = \sqrt{1681} = 41$.
    • $\cot x = \frac{x}{y} = \frac{-40}{-9} = \frac{40}{9}$.

11. Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It is always positive and between $0^\circ$ and $90^\circ$.

  • Quadrant I: Reference angle = Angle

  • Quadrant II: Reference angle = $180^\circ$ - Angle (or $\pi$ - Angle)

  • Quadrant III: Reference angle = Angle - $180^\circ$ (or Angle - $\pi$)

  • Quadrant IV: Reference angle = $360^\circ$ - Angle (or $2\pi$ - Angle)

  • Example: The reference angle for $290^\circ$ (Quadrant IV) is $360^\circ - 290^\circ = 70^\circ$.

  • Example: The reference angle for $210^\circ$ (Quadrant III) is $210^\circ - 180^\circ = 30^\circ$.

12. Evaluating Trigonometric Functions Using Reference Angles

  1. Determine the quadrant of the angle.
  2. Find the reference angle.
  3. Evaluate the trigonometric function for the reference angle (using special triangles or unit circle values).
  4. Apply the correct sign based on the quadrant.
  • Example: $\cos(210^\circ)$:
    1. Quadrant III.
    2. Reference angle is $30^\circ$.
    3. $\cos(30^\circ) = \frac{\sqrt{3}}{2}$.
    4. Cosine is negative in Quadrant III.
    5. Therefore, $\cos(210^\circ) = -\frac{\sqrt{3}}{2}$.

13. Special Angles and Values

  • Unit Circle: Provides exact values for trigonometric functions at common angles (e.g., $30^\circ, 45^\circ, 60^\circ$). Remember $\cos \theta = x$-coordinate, $\sin \theta = y$-coordinate.

  • Special Triangles:

    • 30-60-90: Sides are $1, \sqrt{3}, 2$ (opposite $30^\circ, 60^\circ, 90^\circ$ respectively).
    • 45-45-90: Sides are $1, 1, \sqrt{2}$ (opposite $45^\circ, 45^\circ, 90^\circ$ respectively).

  • Example: $\cos(60^\circ) = \frac{1}{2}$ (adjacent/hypotenuse from 30-60-90 triangle).

  • Example: $\sin(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ (opposite/hypotenuse from 45-45-90 triangle).

  • Example: $\tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ (opposite/adjacent from 30-60-90 triangle).

14. Undefined Trigonometric Values

A trigonometric function is undefined when its denominator is zero.

  • Example: $\tan(90^\circ) = \frac{\sin(90^\circ)}{\cos(90^\circ)} = \frac{1}{0}$, which is undefined.

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