Question

chapter name : trigonometric equations

Solve the given equation...

Answer 1

answer of this question is 45 degree

Answer 2

Answer:

$\implies{\mathsf{\theta = 30^{\circ} \: and \: 45^{\circ} }}$

Step-by-step explanation:

Given , ${\cot}^{2}(\theta) - (1 + \sqrt{3} ) \cot(\theta) + \sqrt{3}=0$

$\implies{\mathsf{0 <\theta< \frac{\pi}{2}}}$

${\cot}^{2}(\theta) - (1 + \sqrt{3} ) \cot(\theta) + \sqrt{3} = 0 \\ \\ { \cot}^{2}(\theta) - \cot(\theta) - \sqrt{3}( \cot(\theta) ) + \sqrt{3} = 0 \\ \\ \cot(\theta) ( \cot(\theta) - 1) - \sqrt{3} ( \cot(\theta) - 1) = 0 \\ \\ ( \cot(\theta) - \sqrt{3})( \cot(\theta) - 1) = 0$

Now equate the factors to zero to get the roots of the equation

$\cot(\theta) - 1 = 0 \\ \\ \cot(\theta) = 1 \\ \\ \cot(\theta) = \cot(\theta)$

$\mathsf{\theta = 45^{\circ}}$

Now equate the next factor,

$\implies{\mathsf{ cot(\theta)- \sqrt{3} =0 }}$

$\implies{\mathsf{ cot(\theta) = \sqrt{3} }}$

$\implies{\mathsf{ \theta = 30^{\circ} }}$

as both 30° & 45° < 90° so are the solutions of the equation.