Mathfolis

Sine, Cosine, and Tangent

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

For an acute angle in a right triangle, sin equals opposite over hypotenuse, cos equals adjacent over hypotenuse, tan equals opposite over adjacent (SOH-CAH-TOA). The unit-circle definition extends these to any angle: for the point (x,y)(x, y) on the unit circle at angle θ\theta, cos⁡θ=x\cos\theta = x, sin⁡θ=y\sin\theta = y, and tan⁡θ=y/x\tan\theta = y/x (where defined).

Definitions

Right triangle
sin⁡=opphyp,    cos⁡=adjhyp,    tan⁡=oppadj\sin = \tfrac{\text{opp}}{\text{hyp}}, \;\; \cos = \tfrac{\text{adj}}{\text{hyp}}, \;\; \tan = \tfrac{\text{opp}}{\text{adj}}
Unit circle
(cos⁡θ,sin⁡θ) on the unit circle at angle θ(\cos\theta, \sin\theta) \text{ on the unit circle at angle } \theta
Quotient identity
tan⁡θ=sin⁡θcos⁡θ\tan\theta = \tfrac{\sin\theta}{\cos\theta}
Pythagorean
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
AP Tip: When a problem gives sin and asks for cos (or vice versa), the Pythagorean identity is faster than reconstructing the triangle.
Type 1

Right-triangle trig ratios

Identify opposite/adjacent/hypotenuse from the labeled angle. Apply SOH-CAH-TOA.

Example 1
In a right triangle, the side opposite angle AA has length 3, the adjacent has length 4, and the hypotenuse has length 5. What is sin⁡A\sin A? (A) 3/43/4 (B) 3/53/5 (C) 4/54/5 (D) 5/35/3

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Type 2

Unit-circle definitions

Identify the x and y coordinates on the unit circle. sin is y; cos is x.

Example 2
The point (32,12)\left(\tfrac{\sqrt{3}}{2}, \tfrac{1}{2}\right) lies on the unit circle at angle θ\theta. What is sin⁡θ\sin\theta? (A) 32\tfrac{\sqrt{3}}{2} (B) 12\tfrac{1}{2} (C) 33\tfrac{\sqrt{3}}{3} (D) 3\sqrt{3}

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Type 3

Find a missing side or angle

Set up a trig equation using the known angle and side. Solve for the unknown side or use inverse trig for the unknown angle.

Example 3
In a right triangle with hypotenuse 10 and one acute angle 30°30°, the opposite side to that angle has length? (A) 55 (B) 535\sqrt{3} (C) 10/310/\sqrt{3} (D) 2020

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