If tan θ = 1/2 and tan ϕ = 1/3, Find the Value of θ + ϕ
Question:
If \[ \tan\theta=\frac{1}{2} \] and \[ \tan\phi=\frac{1}{3} \] then the value of \[ \theta+\phi \] is
If \[ \tan\theta=\frac{1}{2} \] and \[ \tan\phi=\frac{1}{3} \] then the value of \[ \theta+\phi \] is
Solution
Using the tangent addition formula:
\[ \tan(\theta+\phi) = \frac{ \tan\theta+\tan\phi } { 1-\tan\theta\tan\phi } \]
Substituting the given values,
\[ \tan(\theta+\phi) = \frac{ \frac{1}{2}+\frac{1}{3} } { 1-\frac{1}{2}\cdot\frac{1}{3} } \]
\[ = \frac{ \frac{5}{6} } { 1-\frac{1}{6} } \]
\[ = \frac{ \frac{5}{6} } { \frac{5}{6} } \]
\[ =1 \]
Therefore,
\[ \tan(\theta+\phi)=1 \]
Hence,
\[ \theta+\phi=\frac{\pi}{4} \]
Final Answer
\[ \boxed{ \theta+\phi=\frac{\pi}{4} } \]
Correct Option: (d)