Question

Suppose cos(theta) = 5/8 and sin(theta) is < 0. What is the value of cot(theta)?

Answer 1

Given:

$\begin{gathered} \cos \theta=\frac{5}{8} \\ \sin \theta<0 \end{gathered}$

To Find:

$\cot \theta=\text{?}$

Solution:

Please note that

$\cot \theta=\frac{\cos \theta}{\sin \theta}$

For sin(theta) to be less than zero, then the value is negative. From the knowledge of trigonometry, we can get the third side of the triangle from cos(theta) = 5/8, using the Pythagoras theorem.

From the figure above, h would be calculated using the Pythagoras theorem

$\begin{gathered} 8^2=h^2+5^2 \\ 64=h^2+25 \\ h^2=64-25 \\ h^2=39 \\ h=\sqrt[\square]{39} \end{gathered}$

Therefore;

$\text{sin}\theta=\frac{h}{8}=\frac{\sqrt[]{39}}{8}$

Since it is known that sin (theta is negative, then

$\sin \theta=\frac{-\sqrt[]{39}}{8}$

Therefore:

$\begin{gathered} \cot \theta=\frac{\cos\theta}{\text{sin}\theta}=\frac{5}{8}\times-\frac{8}{\sqrt[]{39}} \\ \cot \theta=-\frac{5}{\sqrt[]{39}} \\ \text{Rationalize the denominator would give} \\ \cot \theta=\frac{-5\times\sqrt[]{39}}{\sqrt[]{39}\times\sqrt[]{39}} \\ \cot \theta=\frac{-5\sqrt[]{39}}{39} \end{gathered}$

Hence, cot (theta) is -5√39/39