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: Trig functions convert an angle into a fixed ratio of triangle sides or a coordinate on the unit circle.

Common stuck point: The procedure for trigonometric functions is the easy part; the trap is mixing up which sides belong to sin versus cos. Asking "Am I linking an angle to a ratio of sides (or a point on the unit circle)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I linking an angle to a ratio of sides (or a point on the unit circle)?

Worked Examples

Example 1

easy
Evaluate sin⁡(π6)\sin\left(\frac{\pi}{6}\right) and cos⁡(π6)\cos\left(\frac{\pi}{6}\right).

Answer

sin⁡(π6)=12,cos⁡(π6)=32\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}, \quad \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}

First step

1
Convert the angle mentally: π6\frac{\pi}{6} radians equals 30∘30^\circ.

Full solution

  1. 2
    Recall the special-angle values from the unit circle or a 3030-6060-9090 triangle: sin⁡(30∘)=12\sin(30^\circ) = \frac{1}{2}.
  2. 3
    Using the same reference triangle, cos⁡(30∘)=32\cos(30^\circ) = \frac{\sqrt{3}}{2}.
The special angles (30°,45°,60°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⁡(5π4)\tan\left(\frac{5\pi}{4}\right).

Example 3

medium
In right triangle ABCABC with right angle at CC, AB=13AB = 13, BC=5BC = 5. Find sin⁡A\sin A and cos⁡A\cos A.

Example 4

medium
A ramp 2020 ft long rises 44 ft. Find the angle of elevation θ\theta to the nearest tenth of a degree.

Example 5

hard
From a point 5050 m from the base of a tower, the angle of elevation to the top is 32°32°. Find the tower's height to the nearest meter.

Example 6

challenge
In triangle ABCABC, ∠A=30°\angle A = 30°, ∠B=105°\angle B = 105°, a=8a = 8. Find bb using the Law of Sines.

Practice Problems

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

Example 1

easy
Evaluate cos⁡(π3)\cos\left(\frac{\pi}{3}\right).

Example 2

hard
If sin⁡θ=35\sin\theta = \frac{3}{5} and θ\theta is in Quadrant II, find cos⁡θ\cos\theta and tan⁡θ\tan\theta.

Example 3

easy
Evaluate sin⁡30∘\sin 30^\circ.

Example 4

easy
Evaluate cos⁡60∘\cos 60^\circ.

Example 5

easy
Evaluate tan⁡45∘\tan 45^\circ.

Example 6

easy
In a right triangle, sin⁡θ=opposite?\sin\theta=\frac{\text{opposite}}{?}.

Example 7

easy
Evaluate sin⁡0∘\sin 0^\circ.

Example 8

easy
Evaluate cos⁡0∘\cos 0^\circ.

Example 9

easy
What is the range of sin⁡x\sin x?

Example 10

easy
Evaluate sin⁡90∘\sin 90^\circ.

Example 11

medium
Evaluate sin⁡60∘+cos⁡30∘\sin 60^\circ+\cos 30^\circ.

Example 12

medium
If sin⁡θ=35\sin\theta=\frac{3}{5} and θ\theta is acute, find cos⁡θ\cos\theta.

Example 13

medium
Convert 180∘180^\circ to radians.

Example 14

medium
Find the period of f(x)=sin⁡(2x)f(x)=\sin(2x).

Example 15

medium
Evaluate tan⁡60∘\tan 60^\circ.

Example 16

medium
Find the amplitude and midline of f(x)=3sin⁡x+2f(x)=3\sin x+2.

Example 17

medium
Solve sin⁡x=12\sin x=\frac12 for 0∘≤x<360∘0^\circ\le x<360^\circ.

Example 18

medium
If cos⁡θ=513\cos\theta=\frac{5}{13} and θ\theta acute, find tan⁡θ\tan\theta.

Example 19

challenge
Find the maximum value of f(x)=4sin⁡x−3cos⁡xf(x)=4\sin x-3\cos x.

Example 20

challenge
Solve 2sin⁡2x−sin⁡x−1=02\sin^2 x-\sin x-1=0 for 0∘≤x<360∘0^\circ\le x<360^\circ.

Example 21

challenge
A point starts at (1,0)(1,0) and rotates 120∘120^\circ counterclockwise on the unit circle. Find its coordinates.

Example 22

medium
Find the period of f(x)=cos⁡(x2)f(x)=\cos\left(\frac{x}{2}\right).

Example 23

easy
Evaluate sin⁡45°\sin 45°.

Example 24

easy
Evaluate cos⁡90°\cos 90°.

Example 25

easy
In a right triangle with legs 33 and 44 and hypotenuse 55, sin⁡θ\sin\theta for the angle opposite the side of length 33 is what?

Example 26

easy
Convert π4\dfrac{\pi}{4} radians to degrees.

Example 27

easy
What is the range of cos⁡x\cos x?

Example 28

medium
Evaluate sin⁡ ⁣(5π6)\sin\!\left(\dfrac{5\pi}{6}\right).

Example 29

medium
If cos⁡θ=−32\cos\theta = -\tfrac{\sqrt{3}}{2} and θ\theta is in QIII, find sin⁡θ\sin\theta.

Example 30

medium
Convert 210°210° to radians.

Example 31

medium
Find tan⁡225°\tan 225°.

Example 32

medium
Evaluate cos⁡(−60°)\cos(-60°).

Example 33

medium
Evaluate sin⁡(−45°)\sin(-45°).

Example 34

medium
Solve cos⁡x=0\cos x = 0 for 0≤x<2π0 \le x < 2\pi.

Example 35

medium
State the sign of tan⁡θ\tan\theta in Quadrant II.

Example 36

hard
If tan⁡θ=34\tan\theta = \tfrac{3}{4} and θ\theta is in QIII, find sin⁡θ\sin\theta and cos⁡θ\cos\theta.

Example 37

hard
Solve 2cos⁡x+1=02\cos x + 1 = 0 on [0,2π)[0, 2\pi).

Example 38

hard
Solve sin⁡x=cos⁡x\sin x = \cos x on [0,2π)[0, 2\pi).

Example 39

hard
In triangle ABCABC, a=7a = 7, b=9b = 9, ∠C=60°\angle C = 60°. Find side cc using the Law of Cosines.

Example 40

challenge
Find all x∈[0,2π)x \in [0, 2\pi) with sin⁡x=−32\sin x = -\tfrac{\sqrt{3}}{2}.
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Background Knowledge

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

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