Question

tan theta = 2+ root 3. Find Theta

Answer 1

Answer:

Step-by-step explanation:

2 + $\sqrt{3}$ = $\frac{1 + \sqrt{3} }{1 - \sqrt{3}}$ = $\frac{tan(\pi /4) + tan (\pi /6)}{1 - tan(\pi/4)tan(\pi/6)}$ = tan( $\frac{\pi}{4}$ + $\frac{\pi}{6}$) = tan ( $\frac{5\pi}{12}$ )

therefore, $\theta$ = $\frac{5\pi}{12}$ (Hence proved)

Answer 2

Answer:

Required value of theta is

$\frac{5\pi}{12}$

Step-by-step explanation:

Given,

$\tan(\theta) = 2 + \sqrt{3}$

Now,

$2 + \sqrt{3} = \frac{1 + \sqrt{3} }{1 - \sqrt{3} } \\ = \frac{1 + \sqrt{3} }{1 - \sqrt{3} } \\ = \frac{1 + \sqrt{3} }{1 - 1 \times \sqrt{3} } \\ = \frac{ tan\frac{\pi}{4} + tan\frac{\pi}{6} }{1 -tan \frac{\pi}{4} \times tan \frac{\pi}{6} } \\ = \tan( \frac{\pi}{4} + \frac{\pi}{6} ) \\ = \tan( \frac{(3 + 2)\pi}{12} ) \\ = \tan( \frac{5\pi}{12} )$

So,

$\tan( \theta) = \tan( \frac{5\pi}{12} ) \\ \theta = \frac{5\pi}{12}$

Therefore required value of theta is $\frac{5\pi}{12}$

Here applied formula is $\tan(x + y) = \frac{ \tan(x) + \tan(y) }{1 - \tan(x) \tan(y) }$

Some important Trigonometry formulas,

$sin(x) = \cos(\frac{\pi}{2} - x) \\ \tan(x) = \cot(\frac{\pi}{2} - x) \\ \sec(x) = \csc(\frac{\pi}{2} - x) \\ \cos(x) = \sin(\frac{\pi}{2} - x) \\ \cot(x) = \tan(\frac{\pi}{2} - x) \\ \csc(x) = \sec(\frac{\pi}{2} - x)$

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