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

Write out the given information

`\begin{gathered} \sin x+\cos x=(5)/(4) \\ \\ \sin 2x=2\sin x\cos x \end{gathered}`

From the expression given

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

Then, we have

`\begin{gathered} \sin ^2x+2\sin x\cos x+\cos ^2x=(25)/(16) \\ \text{rearrange } \\ \sin ^2x+\cos ^2x+2\sin x\cos x=(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=(25)/(16)`

Subtract 1 from both sides

`\begin{gathered} 1-1+\sin 2x=(25)/(16)-1 \\ \text{Then} \\ \sin 2x=(9)/(16) \end{gathered}`

Hence

`\begin{gathered} \sin 2x=(p)/(q)=(9)/(16) \\ \\ \text{where } \\ p=9,q=16 \end{gathered}`

Therefore for pq, we have

`pq=p* q=9*16=144`

Hence

pq=144