Question

Suppose sin(theta) = -1/3 and cos(theta) < 0. Use a trig identity to find the value of tan(theta).

Answer 1

Given that;

$\sin \theta-\frac{1}{3},\text{ and cos}\theta<0$

To find the value of tan theta, we shall begin by applying the pythagorean identity as follows;

$\sin ^2\theta+cos^2\theta=1$

We can now substitute for the given value as follows;

$\begin{gathered} (-\frac{1}{3})^2+\cos ^2\theta=1 \\ \frac{1}{9}+\cos ^2\theta=1 \\ Subtract\text{ }\frac{1}{9}\text{ from both sides} \\ \cos ^2\theta=1-\frac{1}{9} \\ \cos ^2\theta=\frac{8}{9} \\ Take\text{ the square root of both sides; } \\ \cos \theta=\sqrt[]{\frac{8}{9}} \\ \cos \theta=\frac{2\sqrt[]{2}}{3} \end{gathered}$

Note however that cos (theta) is less than 0 in the third quadrant, which means

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