Question

If sinA = 4/5 with A in Q2, and sinB = -3/5 with B in Q4, find cos(A - B)

Answer 1

Angle A is in the second quadrant, hence cosA will be negative.

$\begin{gathered} \cos A=-\sqrt[]{1-\sin ^2A} \\ =-\sqrt[]{1-(\frac{4}{5})^2} \\ =\frac{-3}{5} \end{gathered}$

Angle B is in the 4th quadrant, hence cosB will be positive,

$\begin{gathered} \cos B=\sqrt[]{1-\sin ^2B} \\ =\sqrt[]{1-(\frac{-3}{5})^2} \\ =\frac{4}{5} \end{gathered}$

Cos(A-B) can be determined as,

$\begin{gathered} \cos (A-B)=\cos A\cos B+\sin A\sin B \\ =(\frac{-3}{5})\times\frac{4}{5}+\frac{4}{5}\times(\frac{-3}{5}) \\ =\frac{-12}{25}-\frac{12}{25} \\ =\frac{-24}{25} \end{gathered}$

Thus, option (a) is the correct solution.