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:

$\begin{gathered} sin\theta=\frac{opposite}{hypotenuse} \\ sec\theta=\frac{1}{cos\theta}=\frac{hypotenuse}{adjacent} \\ tan\theta=\frac{opposite}{adjacent} \end{gathered}$

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;

$\begin{gathered} sin\theta=\frac{7.55}{11}=0.68 \\ sec\theta=\frac{11}{8}=1.37 \\ tan\theta=\frac{7.55}{8}=1.37 \end{gathered}$