Question

If sin x + cos x = 5/4 and sin (2x) in simplest terms is p/q, compute pq.Note: sin (2x) = 2 sin x cos x

Answer 1

SOLUTION

Write out the given information

$\begin{gathered} \sin x+\cos x=\frac{5}{4} \\ \\ \sin 2x=2\sin x\cos x \end{gathered}$

From the expression given

$\begin{gathered} \sin x+\cos x=\frac{5}{4} \\ \text{Square both sides } \\ (\sin x+\cos x)^2=(\frac{5}{4})^2 \end{gathered}$

Then, we have

$\begin{gathered} \sin ^2x+2\sin x\cos x+\cos ^2x=\frac{25}{16} \\ \text{rearrange } \\ \sin ^2x+\cos ^2x+2\sin x\cos x=\frac{25}{16} \end{gathered}$

Recall that

$\begin{gathered} \sin ^2x+\cos ^2x=1 \\ \text{and} \\ \sin 2x=2\sin x\cos x \\ \end{gathered}$

Substitute into the expression above

$1+\sin 2x=\frac{25}{16}$

Subtract 1 from both sides

$\begin{gathered} 1-1+\sin 2x=\frac{25}{16}-1 \\ \text{Then} \\ \sin 2x=\frac{9}{16} \end{gathered}$

Hence

$\begin{gathered} \sin 2x=\frac{p}{q}=\frac{9}{16} \\ \\ \text{where } \\ p=9,q=16 \end{gathered}$

Therefore for pq, we have

$pq=p\times q=9\times16=144$

Hence

pq=144