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.

The Formula

sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, tan⁡θ=oppositeadjacent=sin⁡θcos⁡θ\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\sin(30°) = 0.5 (opposite/hypotenuse).
cos⁡(60°)=0.5\cos(60°) = 0.5.
tan⁡(45°)=1\tan(45°) = 1.

Notation

sin⁡\sin, cos⁡\cos, tan⁡\tan (and reciprocals csc⁡\csc, sec⁡\sec, cot⁡\cot). Argument in degrees or radians: sin⁡(30°)=sin⁡π6\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⁡θ=opphyp,  cos⁡θ=adjhyp,  tan⁡θ=sin⁡θcos⁡θ\sin\theta = \frac{\text{opp}}{\text{hyp}},\; \cos\theta = \frac{\text{adj}}{\text{hyp}},\; \tan\theta = \frac{\sin\theta}{\cos\theta}; equivalently (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) is the point at angle θ\theta 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.

Common Mistakes

  • Mixing up which sides belong to sin versus cos - SOH-CAH-TOA: sin uses opposite, cos uses adjacent, both over hypotenuse.
  • Leaving the calculator in the wrong angle mode - match degrees or radians to the problem before computing.
  • Using right-triangle ratios for an obtuse or non-right triangle - those need the law of sines or cosines instead.

Why This Formula Matters

Trig is how angles get turned into lengths (and back), powering surveying, navigation, physics of waves, and all periodic modeling. Mixing up which sides go with sin versus cos gives confidently wrong distances and angles. Recognizing it by "Am I linking an angle to a ratio of sides (or a point on the unit circle)?" — rather than by familiar numbers — is what lets a student tell it apart from pythagorean theorem and inverse trig functions and similar triangles in a mixed problem set.

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⁡\sin, cos⁡\cos, tan⁡\tan (and reciprocals csc⁡\csc, sec⁡\sec, cot⁡\cot). Argument in degrees or radians: sin⁡(30°)=sin⁡π6\sin(30°) = \sin\frac{\pi}{6}.

Why is the Trigonometric Functions formula important in Math?

Trig is how angles get turned into lengths (and back), powering surveying, navigation, physics of waves, and all periodic modeling. Mixing up which sides go with sin versus cos gives confidently wrong distances and angles. Recognizing it by "Am I linking an angle to a ratio of sides (or a point on the unit circle)?" — rather than by familiar numbers — is what lets a student tell it apart from pythagorean theorem and inverse trig functions and similar triangles in a mixed problem set.

What do students get wrong about Trigonometric Functions?

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.

What should I learn before the Trigonometric Functions formula?

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

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