Sum and Difference Identities Math Example 4

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Example 4

hard
If sin⁡(α)=45\sin(\alpha) = \frac{4}{5} with α\alpha in QI and cos⁡(β)=−513\cos(\beta) = -\frac{5}{13} with β\beta in QII, find cos⁡(α+β)\cos(\alpha + \beta).

Solution

  1. 1
    Find missing values: cos⁡α=35\cos\alpha = \frac{3}{5} (QI, positive) and sin⁡β=1213\sin\beta = \frac{12}{13} (QII, positive).
  2. 2
    Apply: cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β=35(−513)−45⋅1213=−1565−4865=−6365\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta = \frac{3}{5}\left(-\frac{5}{13}\right) - \frac{4}{5}\cdot\frac{12}{13} = -\frac{15}{65} - \frac{48}{65} = -\frac{63}{65}.

Answer

cos⁡(α+β)=−6365\cos(\alpha + \beta) = -\frac{63}{65}
This problem combines the Pythagorean identity (to find missing trig values), quadrant analysis (to determine signs), and the cosine sum formula. Working with angles in different quadrants requires careful attention to the signs of each trig function.

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.

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More Sum and Difference Identities Examples

Example 1 easy

Find the exact value of [formula] using the sum identity.

Example 2 medium

Simplify [formula].

Example 3 medium

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

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