530,028 views
3 votes
3 votes
Find the exact values of the expression without a calculator

Find the exact values of the expression without a calculator-example-1
User Christian Junk
by
2.0k points

1 Answer

18 votes
18 votes

ANSWER:


(3√(13))/(13)

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}

User Prasoon
by
3.1k points