Question

If $ 7{\sin ^2}\theta + 3{\cos ^2}\theta = 4 $ , where $ \theta $ is an acute angle. Show that $ \cot \theta = \sqrt 3 $ .

Answer 1

Hint: To solve this question, we will rewrite the expression given in the question in a simpler form. Then we will apply the basic trigonometry concept to further resolve this equation. On solving the expression, we will get two values for $ \theta $ as the root of the expression. We will consider the positive root to get the value of $ \cot \theta $ .

Complete step-by-step answer:
Given:
The given trigonometric relation is $ 7{\sin ^2}\theta + 3{\cos ^2}\theta = 4 $ , where $ \theta $ is an acute angle.
We will rewrite the above expression to simplify the problem.
 $ \begin{array}{c}
4{\sin ^2}\theta + 3{\sin ^2}\theta + 3{\cos ^2}\theta = 4\\
4{\sin ^2}\theta + 3\left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right) = 4
\end{array} $
We know from the basics of trigonometry that $ {\sin ^2}\theta + {\cos ^2}\theta = 1 $ . We will apply this relation in the above expression.
 $ 4{\sin ^2}\theta + 3 = 4 $
We will rewrite the above expression to find the value of $ \sin \theta $ .
 $ \begin{array}{c}
4{\sin ^2}\theta = 1\\
{\sin ^2}\theta = \dfrac{1}{4}\\
\sin \theta = \sqrt {\dfrac{1}{4}} \\
\sin \theta = \pm \dfrac{1}{2}
\end{array} $
Therefore, we have $ \sin \theta = \dfrac{1}{2} $ and $ \sin \theta = - \dfrac{1}{2} $ .
On considering the positive root, we will obtain the value of angle,
 $ \begin{array}{c}
\sin \theta = \dfrac{1}{2}\\
\theta = {\sin ^{ - 1}}\left( {\dfrac{1}{2}} \right)\\
\theta = \dfrac{\pi }{6}
\end{array} $
Now we will substitute $ \dfrac{\pi }{6} $ for $ \theta $ to find $ \cot \theta $ .
 $ \begin{array}{c}
\cot \theta = \cot \left( {\dfrac{\pi }{6}} \right)\\
\cot \theta = \sqrt 3
\end{array} $
Therefore, it proved from the above expression that the value of $ \cot \theta $ is equal to $ \sqrt 3 $ .

Note: To solve this question, we will require the prior knowledge of basic trigonometry. As the expression is in the form of both cosine and sine, we will always have to convert the equation in the form of one of these to find the value of $ \theta $ . This approach will simplify the expression. We will always have to consider the roots of the equation according to the condition or results given in the question. This problem can also be solved by converting it into cosine form, but the procedure will be a little complicated.