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Sine and Cosine Functions is taught in the note for Trigonometry. Read it below and look for the section on this topic.

Trigonometry

14 min read

Sine and cosine for any angle

For a point P(x, y) on a circle of radius r at an angle θ measured from the positive x-axis, sin θ = y/r and cos θ = x/r. The values are positive or negative depending on the quadrant. Remember ASTC: All positive in the 1st quadrant, Sine in the 2nd, Tangent in the 3rd, Cosine in the 4th.

Sine, cosine and tangentA right-angled triangle with an angle theta at the left. The side opposite theta is the opposite, the side next to theta that is not the longest is the adjacent, and the longest side is the hypotenuse. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent.θoppositeadjacenthypotenusesin θ = opposite ÷ hypotenusecos θ = adjacent ÷ hypotenusetan θ = opposite ÷ adjacentSOH CAH TOA
Sine rises from 0 to 1 at 90°, falls to −1 at 270° and repeats every 360°.
Cosine has the same shape, starting at its maximum.
θ0°30°45°60°90°180°270°
sin θ01/2√2/2√3/210−1
cos θ1√3/2√2/21/20−10

Compound angles

These formulas give the sine and cosine of the sum or difference of angles.
Worked exampleSin 75°

Find the exact value of sin 75°.

  1. Write 75° = 45° + 30°.
  2. Substitute.

Answer: sin 75° = (√6 + √2) ÷ 4 ≈ 0.966.

Try itCheck the compound-angle formula

Choose two angles. The formula gives the same as sin(A + B) calculated directly.

Check yourself sin 150° is
Check yourself cos 120° is
Check yourself sin(A + B) is

Key terms

Quadrant
One of four regions of the plane.
Compound angle
An angle that is a sum or difference of two angles.
Period
The length after which the curve repeats: 360°.
Amplitude
The maximum value of the graph.

Summary

  • sin θ = y/r and cos θ = x/r for any angle.
  • The signs follow ASTC; both curves repeat every 360°.
  • sin(A ± B) and cos(A ± B) have standard formulas.
  • Exact values come from special triangles.
Compound Angles