Mathfolis

Sine and Cosine Function Values

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

AP Precalc expects fluency with sin and cos at the standard special angles in radians: 0, π/6, π/4, π/3, π/2 — plus their multiples in the other three quadrants. Two reduction techniques cover everything else: subtract multiples of 2π2\pi to find a coterminal angle in [0,2π)[0, 2\pi), and use a reference angle (the acute angle to the x-axis) with the appropriate quadrant sign.

Special angle values

sin⁡0=0,cos⁡0=1\sin 0 = 0, \cos 0 = 1
sin⁡ ⁣π6=12,cos⁡ ⁣π6=32\sin\!\tfrac{\pi}{6} = \tfrac{1}{2}, \cos\!\tfrac{\pi}{6} = \tfrac{\sqrt 3}{2}
sin⁡ ⁣π4=cos⁡ ⁣π4=22\sin\!\tfrac{\pi}{4} = \cos\!\tfrac{\pi}{4} = \tfrac{\sqrt 2}{2}
sin⁡ ⁣π3=32,cos⁡ ⁣π3=12\sin\!\tfrac{\pi}{3} = \tfrac{\sqrt 3}{2}, \cos\!\tfrac{\pi}{3} = \tfrac{1}{2}
sin⁡ ⁣π2=1,cos⁡ ⁣π2=0\sin\!\tfrac{\pi}{2} = 1, \cos\!\tfrac{\pi}{2} = 0

Quadrant signs (ASTC: All, Sin, Tan, Cos positive)

Q1
sin⁡,cos⁡,tan⁡ all >0\sin, \cos, \tan \text{ all } > 0
Q2
sin⁡>0;cos⁡,tan⁡<0\sin > 0; \cos, \tan < 0
Q3
tan⁡>0;sin⁡,cos⁡<0\tan > 0; \sin, \cos < 0
Q4
cos⁡>0;sin⁡,tan⁡<0\cos > 0; \sin, \tan < 0
AP Tip: ASTC: 'All Students Take Calculus' — All in Q1, Sin in Q2, Tan in Q3, Cos in Q4 are the positive functions in each quadrant.
Type 1

Exact value at a special angle

Recall sin or cos at the named special angle directly.

Example 1
What is cos⁡ ⁣π3\cos\!\tfrac{\pi}{3}? (A) 12\tfrac{1}{2} (B) 22\tfrac{\sqrt 2}{2} (C) 32\tfrac{\sqrt 3}{2} (D) 11

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Type 2

Use a reference angle

Find the acute angle to the x-axis, evaluate sin or cos there, then attach the correct sign for the quadrant.

Example 2
What is sin⁡ ⁣5π6\sin\!\tfrac{5\pi}{6}? (A) −12-\tfrac{1}{2} (B) 12\tfrac{1}{2} (C) −32-\tfrac{\sqrt 3}{2} (D) 32\tfrac{\sqrt 3}{2}

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Type 3

Coterminal angles

Subtract or add multiples of 2π2\pi until the angle lies in [0,2π)[0, 2\pi).

Example 3
Reduce 13π4\tfrac{13\pi}{4} to its smallest positive coterminal angle in [0,2π)[0, 2\pi). (A) π4\tfrac{\pi}{4} (B) 3π4\tfrac{3\pi}{4} (C) 5π4\tfrac{5\pi}{4} (D) 7π4\tfrac{7\pi}{4}

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