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Question

Find the value of $\theta :$$\mathrm{tan}\theta =\mathrm{sin}45°\mathrm{cos}45°+\mathrm{sin}30°$

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Solution

Step-1 Trigonometric ratios value:As,$\mathrm{sin}45°=\mathrm{cos}45°=\frac{1}{\sqrt{2}}\phantom{\rule{0ex}{0ex}}\mathrm{sin}30°=\frac{1}{2},\mathrm{tan}45°=1$Step-2 Solving the given expression:In order to determine the value of $\theta$, we will simplify the given expression, $\begin{array}{rcl}\mathrm{tan}\theta & =& \mathrm{sin}45°\mathrm{cos}45°+\mathrm{sin}30°\\ & ⇒& \mathrm{tan}\theta =\frac{1}{\sqrt{2}}×\frac{1}{\sqrt{2}}+\frac{1}{2}\\ & ⇒& \mathrm{tan}\theta =\frac{1}{2}+\frac{1}{2}\\ & ⇒& \mathrm{tan}\theta =1\\ & ⇒& \mathrm{tan}\theta =\mathrm{tan}45°\\ ⇒\theta & =& 45°\end{array}$Hence, the value of $\theta$ in the given expression will be $45°$.

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