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Complementary Trigonometric Ratios

given that ta...

Question

given that tan (theta 1 +theta 2)=tan theta 1 +tan theta 2/ 1-tan theta 1 tan theta 2 . find theta 1 +theta 2 when tan theta 1 =1/2 and tan theta 2 =1/3

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Solution

Answer :

Given : $\tan (θ_{1} + θ_{2}) = \frac{\tan θ_{1} + \tan θ_{2}}{1 - t a n θ_{1} t a n θ_{2}}$ and $t a n θ_{1} = \frac{1}{2}, t a n θ_{2} = \frac{1}{3}$

Substitute these values we get

$\Rightarrow \tan (θ_{1} + θ_{2}) = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \frac{1}{2} \times \frac{1}{3}} \Rightarrow t a n (θ_{1} + θ_{2}) = \frac{\frac{3 + 2}{6}}{1 - \frac{1}{6}} \Rightarrow t a n (θ_{1} + θ_{2}) = \frac{\frac{5}{6}}{\frac{6 - 1}{6}} \Rightarrow t a n (θ_{1} + θ_{2}) = \frac{\frac{5}{6}}{\frac{5}{6}} \Rightarrow t a n (θ_{1} + θ_{2}) = 1 \Rightarrow t a n (θ_{1} + θ_{2}) = \tan 45 ° (W e k n o w t a n 45 ° = 1) \Rightarrow θ_{1} + θ_{2} = 45 ° (A n s)$

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