Find the exact value under the given conditions.
-$\tan \alpha = \frac { 5 } { 12 } , \pi

Question

Find the exact value under the given conditions.
-$\tan \alpha = \frac { 5 } { 12 } , \pi < \alpha < \frac { 3 \pi } { 2 } ; \quad \cos \beta = - \frac { 3 } { 5 } , \frac { \pi } { 2 } < \beta < \pi \quad$ Find $\tan ( \alpha + \beta )$
A) $- \frac { 33 } { 56 }$
B) $\frac { 56 } { 65 }$
C) $\frac { 2 } { 7 }$
D) $- \frac { 33 } { 65 }$

Quizplus · Accepted Answer

The formula for $\tan(\alpha + \beta)$ is $\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$. Given $\tan \alpha = \frac{5}{12}$, and $\cos \beta = -\frac{3}{5}$, we find $\sin \beta$ using $\sin^2 \beta = 1 - \cos^2 \beta$, which gives $\sin \beta = \frac{4}{5}$ (negative because $\frac{\pi}{2} < \beta < \pi$). Thus, $\tan \beta = \frac{\sin \beta}{\cos \beta} = \frac{4}{-3}$. Substituting these into the formula gives $\frac{\frac{5}{12} - \frac{4}{3}}{1 - \frac{5}{12} \cdot \left(-\frac{4}{3}\right)} = -\frac{33}{56}$.

Find the Exact Value Under the Given Conditions Find tan⁡(α+β)\tan ( \alpha + \beta )tan(α+β)

Find the Exact Value Under the Given Conditions Find $\tan ( \alpha + \beta )$