IBHardGeometry & Trigonometry10–12

Exact Sine and Cosine of a Sum of Two Angles

Seno e Cosseno Exatos da Soma de Dois Ângulos

The angles $A$ and $B$ are acute, with $\sin A = \frac{3}{5}$ and $\cos B = \frac{5}{13}$ . Find the exact values of $\sin(A + B)$ and $\cos(A + B)$ , and state what the sign of $\cos(A + B)$ says about the angle $A + B$ .

Each compound angle formula needs four values and you are given two, so recover the missing sine and cosine from the Pythagorean identity, choosing the sign from the quadrant the angles lie in.

Solution

Step 1 of 12

1.Both angles are acute, so both cosines and both sines are positive.