Mathematics.

trigonometric identities

Sum and Difference Formulas

Trigonometry25 minDifficulty3 out of 10

You should know: trigonometric identities

Overview

The sum and difference formulas express the sine, cosine, and tangent of a sum or difference of two angles in terms of the sines, cosines, and tangents of the individual angles. They let you compute exact trigonometric values for angles like 15° or 75° that are not standard angles themselves but can be written as sums or differences of standard angles such as 30°, 45°, and 60°. These formulas are foundational: nearly every other trigonometric identity — double-angle, half-angle, product-to-sum — can be derived from them.

Intuition

Think of adding two rotations: rotating a point first by angle A and then by angle B should give the same result as rotating it once by A+B. Writing that composition of rotations out in coordinates — using the rotation matrix for angle A applied to the point already rotated by B — produces exactly the sum formulas. This is why the formulas mix sine and cosine of both angles: each new coordinate is a blend of the old x- and y-components.

Formal Definition

Definition

For any angles A and B:

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B
Sine sum/difference
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B
Cosine sum/difference
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}
Tangent sum/difference

Worked Examples

  1. Apply the sine sum formula with A = 45°, B = 30°.

    sin⁡(75°)=sin⁡45°cos⁡30°+cos⁡45°sin⁡30°\sin(75°) = \sin45°\cos30° + \cos45°\sin30°

  2. Substitute the known exact values.

    =22⋅32+22⋅12=6+24= \frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2} + \frac{\sqrt2}{2}\cdot\frac{1}{2} = \frac{\sqrt6+\sqrt2}{4}

Answer: sin(75°) = (√6 + √2)/4 ≈ 0.9659

Practice Problems

Difficulty 3/10

Find the exact value of cos(15°) using 15° = 45° − 30°.

Difficulty 4/10

If sin A = 3/5 (A in quadrant I) and cos B = 5/13 (B in quadrant I), find sin(A+B).

Difficulty 6/10

Two sound waves with the same frequency combine as sin(ωt) + sin(ωt + 60°). Use the sum formula to write cos60° and sin60° contributions and show the combined amplitude is not simply 2.

Quiz

cos(A − B) equals:
sin(75°), computed as sin(45°+30°), equals:
The sum and difference formulas are foundational because:

Summary

  • sin(A±B) = sinA cosB ± cosA sinB and cos(A±B) = cosA cosB ∓ sinA sinB.
  • They let you compute exact values at non-standard angles (like 15° or 75°) by writing them as sums/differences of standard angles.
  • They are the derivation source for double-angle, half-angle, and product-to-sum identities.

References

  1. WebsiteWikipedia — List of trigonometric identities