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Trigonometric Functions — Definition, Formula & Examples

Trigonometric functions are functions that relate an angle of a right triangle to ratios of two of its sides. The six standard trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.

For an acute angle θ\theta in a right triangle, the trigonometric functions are defined as ratios of the triangle's sides: sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, and tan⁡θ=oppositeadjacent\tan\theta = \frac{\text{opposite}}{\text{adjacent}}. These definitions extend to all real-valued angles via the unit circle, where sin⁡θ\sin\theta and cos⁡θ\cos\theta correspond to the yy- and xx-coordinates of a point on the circle.

Key Formula

sin⁡θ=opphyp,cos⁡θ=adjhyp,tan⁡θ=oppadj\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}
Where:
  • θ\theta = The angle of interest in the triangle
  • opp\text{opp} = Length of the side opposite the angle
  • adj\text{adj} = Length of the side adjacent to the angle (not the hypotenuse)
  • hyp\text{hyp} = Length of the hypotenuse

How It Works

Pick an angle θ\theta in a right triangle. Label the side across from θ\theta as "opposite," the side next to θ\theta (that is not the hypotenuse) as "adjacent," and the longest side as the "hypotenuse." Form the ratio that matches whichever trig function you need. The mnemonic SOH-CAH-TOA helps: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. The remaining three functions are reciprocals: csc⁡θ=1/sin⁡θ\csc\theta = 1/\sin\theta, sec⁡θ=1/cos⁡θ\sec\theta = 1/\cos\theta, and cot⁡θ=1/tan⁡θ\cot\theta = 1/\tan\theta.

Worked Example

Problem: A right triangle has an angle θ, an opposite side of length 3, an adjacent side of length 4, and a hypotenuse of length 5. Find sin θ, cos θ, and tan θ.
Find sin θ: Divide the opposite side by the hypotenuse.
sin⁡θ=35=0.6\sin\theta = \frac{3}{5} = 0.6
Find cos θ: Divide the adjacent side by the hypotenuse.
cos⁡θ=45=0.8\cos\theta = \frac{4}{5} = 0.8
Find tan θ: Divide the opposite side by the adjacent side.
tan⁡θ=34=0.75\tan\theta = \frac{3}{4} = 0.75
Answer: sin⁡θ=0.6\sin\theta = 0.6, cos⁡θ=0.8\cos\theta = 0.8, tan⁡θ=0.75\tan\theta = 0.75.

Why It Matters

Trigonometric functions appear throughout physics, engineering, and calculus — from modeling sound waves to computing forces on a bridge. In precalculus and AP Calculus, you will differentiate and integrate these functions regularly, so fluency with their definitions and values is essential.

Common Mistakes

Mistake: Mixing up which side is "opposite" and which is "adjacent" when the angle changes.
Correction: Always label sides relative to the specific angle you are working with. The opposite and adjacent sides swap if you switch to the other acute angle in the triangle.