Question

Find the exact valuse of the expression sin( cos^-1 (5/13))

Answer 1

Given:

`sin(cos^(-1)((5)/(13)))`

To Determine: The exact value of the given expression

Solution

Let

`\begin{gathered} cos^(-1)((5)/(13))=x \\ cosx=(5)/(13) \end{gathered}`

Also note that

`\begin{gathered} sin^2x+cos^2x=1 \\ sin^2x=1-cos^2x \\ sinx=√(1-cos^2x) \\ Therefore \\ sin(cos^(-1)((5)/(13))=sinx \\ sinx=\sqrt{1-((5)/(13))^2} \end{gathered}`
`\begin{gathered} sinx=\sqrt{1-(25)/(169)} \\ sinx=\sqrt{(169-25)/(169)} \\ sinx=\sqrt{(144)/(169)} \\ sinx=(12)/(13) \end{gathered}`

Hence

`sin(cos^(-1)((5)/(13)))=(12)/(13)`