Question

If (sin theta + cos theta)^2 =5/2 then sin theta cos theta =?

Answer 1

Given:

`(\text{sin}\theta+\cos \theta)^2\text{ = }(5)/(2)`

Let's solve for sin θ cosθ

First expand the parenthesis:

`\begin{gathered} (\sin \theta+\cos \theta)^2\text{ = }(5)/(2) \\ \\ (\sin \theta+\cos \theta)(\sin \theta+\cos \theta)\text{ = }(5)/(2) \\ \\ \sin ^2\theta+\sin \theta cos\theta+\sin \theta\cos \theta+\cos ^2\theta\text{ = }(5)/(2) \\ We\text{ know: +}\cos ^2\theta+sin^2\theta=1 \\ ^{}\sin \theta cos\theta+\sin \theta\cos \theta\text{ +}\cos ^2\theta+sin^2\theta\text{ = }(5)/(2) \\ 2\sin \theta cos\theta+1\text{ = }(5)/(2) \end{gathered}`

`\begin{gathered} \\ 2\sin \theta\cos \theta\text{ + 1 = }(5)/(2) \\ \\ 2\sin \theta\cos \theta\text{ = }(5)/(2)-1 \\ \\ 2\sin \theta\cos \theta=\text{ }(3)/(2) \\ \\ \sin \theta\cos \theta\text{ = }(3)/(2)\ast(1)/(2) \\ \\ \sin \theta\cos \theta\text{ = }(3)/(4) \\ \\ \end{gathered}`

ANSWER:

`(3)/(4)`