Question

find the acute angle theta when

cos theta - sin theta / cos theta + sin theta = 1 - root3 / 1 + root3

Answer 1

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MARK AS BRAINLIEST

Answer 2

$\textbf{ \underline{ \underline{QUESTION}}}$

$\text{Find an acute angle} \: {\theta }.... \\ \\ \text{when } \: \frac{ \text{cos} \: \theta - \text{sin} \: \theta}{\text{cos} \: \theta + \text{sin} \: \theta} = \frac{ 1 - \sqrt{3} }{1 - \sqrt{3} }$

$\textbf{ \underline{ \underline{SOLUTION}}}$

$\text{we \: have....} \\ \\ \frac{ \text{cos} \: \theta - \text{sin} \: \theta}{\text{cos} \: \theta + \text{sin} \: \theta} = \frac{ 1 - \sqrt{3} }{1 - \sqrt{3} } \\ \\ \implies \frac{( \text{cos} \theta - \text{sin} \theta) + ( \text{cos} \theta + \text{sin} \theta)}{( \text{cos} \theta - \text{sin} \theta) - ( \text{cos} \theta + \text{sin} \theta)} = \frac{(1 - \sqrt{3} ) + (1 + \sqrt{3} )}{(1 - \sqrt{3} ) - (1 + \sqrt{3} )} \: \: \: \: [\textsf {Applying componendo and dividendo}]\\ \\ \implies {\frac{\text {cos} \theta - \text{sin} \theta + \text{cos} \theta + \text{sin} \theta}{\text{cos} \theta - \text{sin} \theta - \text{cos} \theta - \text{sin} \theta} }= \frac{1 - \sqrt{3 } + 1 + \sqrt{3} }{1 - \sqrt{3} - 1 - \sqrt{3} } \\ \\ \implies \frac{ \text{2\:cos} \theta}{ \text{ - 2\:sin} \theta} = \frac{2}{ - 2 \sqrt{3} } \\ \\ \implies \frac{ \text{cos} \theta}{ \text{sin} \theta} = \frac{1}{ \sqrt{3} } \: \: \: \: \: \: \: \: \: \:[\textsf {2 canceled 2}]\\ \\ \implies \text{cot} \theta = \frac{1}{ \sqrt{3} } \: \: \: \: \: \: \: \: \: [\frac {\textsf {cos}\theta}{\textsf {sin}\theta} = \textsf {cot}\theta]\\ \\ \implies \frac{1}{ \text{tan} \theta} = \frac{1}{ \sqrt{3} }\: \: \: \: \:\: \: \: \: \:[\text {cot}\theta = \frac {1}{\textsf {tan}\theta}]\\ \\ \implies \text{tan} \theta = \sqrt{3} \\ \\ \implies \text{tan} \theta = \text{tan60}\degree \: \: \:\:[\textsf{on comparing two sides}] \\ \\ \implies \theta = 60\degree$

$\textsf{so \: the \: acute \: angle } \theta \: \text{is} \: 60\degree$