Question

Find sin theta, sec theta, and tan theta, where theta is the angle shown in the figure. Give exact values, not decimal approximations.

Answer 1

Based on the definition of sin, sec and tan of an angle, you have:

`\begin{gathered} sin\theta=(opposite)/(hypotenuse) \\ sec\theta=(1)/(cos\theta)=(hypotenuse)/(adjacent) \\ tan\theta=(opposite)/(adjacent) \end{gathered}`

the opposite side length can be calculated by using the Pythagorean theorem:

`\begin{gathered} h^2=c_1^2+c_2^2 \\ c_1^2=h^2-c_2^2 \\ c_1=\sqrt[\placeholder{⬚}]{11^2-8^2} \\ c_1\approx7.55 \end{gathered}`

then, by replacing the values of the lengths of hypotenuse, opposite and adjacent sides, you obtain;

`\begin{gathered} sin\theta=(7.55)/(11)=0.68 \\ sec\theta=(11)/(8)=1.37 \\ tan\theta=(7.55)/(8)=1.37 \end{gathered}`