Trigonometric Functions Formula

The Formula

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}

When to use: Angles have numbers associated with them—sin, cos, tan capture different ratios.

Quick Example

\sin(30°) = 0.5 (opposite/hypotenuse).
\cos(60°) = 0.5.
\tan(45°) = 1.

Notation

\sin, \cos, \tan (and reciprocals \csc, \sec, \cot). Argument in degrees or radians: \sin(30°) = \sin\frac{\pi}{6}.

What This Formula Means

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.

Formal View

\sin\theta = \frac{\text{opp}}{\text{hyp}},\; \cos\theta = \frac{\text{adj}}{\text{hyp}},\; \tan\theta = \frac{\sin\theta}{\cos\theta}; equivalently (\cos\theta, \sin\theta) is the point at angle \theta on the unit circle

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).

Common Mistakes

  • Mixing up SOH-CAH-TOA — sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, tangent is opposite/adjacent
  • Using degree values in formulas that expect radians — \sin(90) on a radian-mode calculator gives 0.894, not 1
  • Forgetting that trig functions have specific ranges — \sin and \cos are bounded between -1 and 1

Why This Formula Matters

Model anything periodic: sound, light, seasons, electronics.

Frequently Asked Questions

What is the Trigonometric Functions formula?

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.

How do you use the Trigonometric Functions formula?

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

What do the symbols mean in the Trigonometric Functions formula?

\sin, \cos, \tan (and reciprocals \csc, \sec, \cot). Argument in degrees or radians: \sin(30°) = \sin\frac{\pi}{6}.

Why is the Trigonometric Functions formula important in Math?

Model anything periodic: sound, light, seasons, electronics.

What do students get wrong about Trigonometric Functions?

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

What should I learn before the Trigonometric Functions formula?

Before studying the Trigonometric Functions formula, you should understand: triangles, ratios.

← Back to Trigonometric Functions See All Examples Practice Problems