Question

Find the exact value of sin(tan^-1(-5/12))

Answer 1

we have the expression

`sin\left(tan^(-1)\left(-5/12\right)\right)`

Let

x ----> the measure of the given angle in degrees

we have that

`\begin{gathered} tan(x)=-(5)/(12) \\ so \\ sin(tan^(-1)(-(5)/(12))=sinx \\ \end{gathered}`

Remember the identity

`\begin{gathered} tan^2x+1=sec^2x \\ substitute\text{ given value} \\ If\text{ the value of the tangent is negative the angle x lies on the II quadrant or IV quadrant} \\ (-(5)/(12))^2+1=sec^2x \\ sec^2x=(25)/(144)+1 \\ \\ sec^2x=(169)/(144) \\ \\ secx=\pm(13)/(12) \\ \\ cosx=\pm(12)/(13) \end{gathered}`

Find out the value of sine of the angle x

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