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=(5)/(8) \\ \sin \theta<0 \end{gathered}`

To Find:

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

Solution:

Please note that

`\cot \theta=(\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=(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}=(5)/(8)*-\frac{8}{\sqrt[]{39}} \\ \cot \theta=-\frac{5}{\sqrt[]{39}} \\ \text{Rationalize the denominator would give} \\ \cot \theta=\frac{-5*\sqrt[]{39}}{\sqrt[]{39}*\sqrt[]{39}} \\ \cot \theta=\frac{-5\sqrt[]{39}}{39} \end{gathered}`

Hence, cot (theta) is -5√39/39