Question

Find the value of $\theta :$

$\sqrt{3}\tan 2\theta =\cos 60°+\sin 45°\cos 45°$

Answer 1

Step-1Trigonometric ratios value:

As,

$$\begin{gathered} \cos 60°=\frac{1}{2},\sin 45°=\frac{1}{\sqrt{2}}=\cos 45° \\ \tan 30°=\frac{1}{\sqrt{3}} \end{gathered}$$

Step-2 Solving the given expression:

In order to determine the value of $\theta$, we will simplify the given expression,

$\begin{array}{rcl}\sqrt{3}\tan 2\theta & =& \cos 60°+\sin 45°\cos 45°\\ & \Rightarrow & \sqrt{3}\tan 2\theta =\frac{1}{2}+\frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}}\\ & \Rightarrow & \sqrt{3}\tan 2\theta =\frac{1}{2}+\frac{1}{2}\\ & \Rightarrow & \sqrt{3}\tan 2\theta =1\\ \Rightarrow \tan 2\theta & =& \frac{1}{\sqrt{3}}\\ \Rightarrow \tan 2\theta & =& \tan 30°\\ \Rightarrow 2\theta & =& 30°\\ \Rightarrow \theta & =& \frac{30}{2}=15°\end{array}$

Hence, the value of $\theta$ in the given expression will be $15°$.