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