Question 1

Find the exact values of the expression without a calculator

Question 2

ANSWER:

Explanation:

We have the following expression:

$\cos\left(\tan^(-1)\left(-(2)/(3)\right)\right)$

$\begin{gathered} \cos\left(\tan^(-1)\left(-(2)/(3)\right)\right)=\cos\left(\arctan\left(-(2)/(3)\right)\right) \\ \\ \cos\left(-\arctan\left((2)/(3)\right)\right)=\cos\left(\arctan\left((2)/(3)\right)\right) \\ \\ \text{ we have the following} \\ \\ \cos \left(\arctan \left(x\right)\right)=(√(1+x^2))/(1+x^2) \\ \\ \text{ we replacing:} \\ \\ \cos\left(\arctan\left((2)/(3)\right)\right)=\frac{\sqrt{1+\left((2)/(3)\right)^2}}{1+\left((2)/(3)\right)^2}=\frac{\sqrt{1+\left((2)/(3)\right)^2}}{1+(2^2)/(3^2)}=\frac{\sqrt{1+(4)/(9)}}{1+(4)/(9)}=\frac{\sqrt{(13)/(9)}}{(13)/(9)}=((1)/(3)√(13))/((13)/(9))=(√(13))/(3\cdot(13)/(9))=(√(13))/((13)/(3))=(3√(13))/(13) \end{gathered}$

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