Sum and Difference Identities Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Find the exact value of cos⁡(75°)\cos(75°) using the sum identity.

Solution

  1. 1
    Write 75°=45°+30°75° = 45° + 30°.
  2. 2
    Apply the cosine sum formula: cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B) = \cos A \cos B - \sin A \sin B.
  3. 3
    Substitute: cos⁡(75°)=cos⁡(45°)cos⁡(30°)−sin⁡(45°)sin⁡(30°)\cos(75°) = \cos(45°)\cos(30°) - \sin(45°)\sin(30°).
  4. 4
    =22⋅32−22⋅12=64−24=6−24= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}.

Answer

cos⁡(75°)=6−24\cos(75°) = \frac{\sqrt{6} - \sqrt{2}}{4}
The sum and difference identities let us find exact values for angles that are sums or differences of standard angles (30°, 45°, 60°). Here we decomposed 75° as 45° + 30° and applied the cosine addition formula.

About Sum and Difference Identities

Formulas that express sin⁡(A±B)\sin(A \pm B), cos⁡(A±B)\cos(A \pm B), and tan⁡(A±B)\tan(A \pm B) in terms of sin⁡A\sin A, cos⁡A\cos A, sin⁡B\sin B, and cos⁡B\cos B.

Learn more about Sum and Difference Identities →

More Sum and Difference Identities Examples

Example 2 medium

Simplify [formula].

Example 3 medium

Find the exact value of [formula] using a difference identity.

Example 4 hard

If [formula] with [formula] in QI and [formula] with [formula] in QII, find [formula].

← All Sum and Difference Identities Examples Sum and Difference Identities Concept