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

User Pockets
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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

Suppose cos(theta) = 5/8 and sin(theta) is < 0. What is the value of cot(theta-example-1
User Rik Hemsley
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7.9k points

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