Trigonometric Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Trigonometric Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Trigonometric functions (sin, cos, tan, etc.) relate angles in right triangles to side ratios and extend to periodic functions of real numbers via the unit circle.

Angles have numbers associated with them—sin, cos, tan capture different ratios.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: These functions extend beyond triangles to describe circular motion and waves.

Common stuck point: Radians vs. degrees: \pi radians = 180°. Most calculators default to degrees.

Sense of Study hint: Draw a right triangle, label the sides opposite, adjacent, and hypotenuse relative to your angle, then use SOH-CAH-TOA.

Worked Examples

Example 1

easy
Evaluate \sin\left(\frac{\pi}{6}\right) and \cos\left(\frac{\pi}{6}\right).

Solution

  1. 1
    Convert the angle mentally: \frac{\pi}{6} radians equals 30^\circ.
  2. 2
    Recall the special-angle values from the unit circle or a 30-60-90 triangle: \sin(30^\circ) = \frac{1}{2}.
  3. 3
    Using the same reference triangle, \cos(30^\circ) = \frac{\sqrt{3}}{2}.

Answer

\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}, \quad \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
The special angles (30°, 45°, 60°) and their radian equivalents appear frequently. Memorizing the unit circle values or using the 30-60-90 and 45-45-90 triangle ratios is essential.

Example 2

medium
Find the exact value of \tan\left(\frac{5\pi}{4}\right).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Evaluate \cos\left(\frac{\pi}{3}\right).

Example 2

hard
If \sin\theta = \frac{3}{5} and \theta is in Quadrant II, find \cos\theta and \tan\theta.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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