Find the exact value under the given conditions.
-$\cos \alpha = - \frac { 5 } { 13 } , \frac { \pi } { 2 }

Question

Find the exact value under the given conditions.
-$\cos \alpha = - \frac { 5 } { 13 } , \frac { \pi } { 2 } < \alpha < \pi ; \quad \sin \beta = \frac { 8 } { 17 } , \frac { \pi } { 2 } < \alpha < \pi \quad$ Find $\tan ( \alpha - \beta )$.
A) $\frac { 50 } { 19 }$

B) $\frac { 20 } { 3 }$

C) $- \frac { 20 } { 3 }$

D) $- \frac { 140 } { 171 }$

Quizplus · Accepted Answer

Given $\cos \alpha = -\frac{5}{13}$ and $\sin \beta = \frac{8}{17}$, we find the sine and cosine of $\beta$ and the sine of $\alpha$ to use the formula for $\tan(\alpha - \beta)$. Since $\frac{\pi}{2} < \alpha < \pi$ and $\frac{\pi}{2} < \alpha < \pi$, both $\alpha$ and $\beta$ are in the second quadrant where sine is positive and cosine is negative. For $\alpha$, $\sin \alpha = \sqrt{1 - \cos^2 \alpha} = \sqrt{1 - \left(-\frac{5}{13}\right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13}$. For $\beta$, $\cos \beta = -\sqrt{1 - \sin^2 \beta} = -\sqrt{1 - \left(\frac{8}{17}\right)^2} = -\sqrt{1 - \frac{64}{289}} = -\sqrt{\frac{225}{289}} = -\frac{15}{17}$. Using the formula for $\tan(\alpha - \beta) = \frac{\sin \alpha \cos \beta - \cos \alpha \sin \beta}{\cos \alpha \cos \beta + \sin \alpha \sin \beta}$, we get $\tan(\alpha - \beta) = \frac{\frac{12}{13} \cdot -\frac{15}{17} - -\frac{5}{13} \cdot \frac{8}{17}}{-\frac{5}{13} \cdot -\frac{15}{17} + \frac{12}{13} \cdot \frac{8}{17}} = \frac{-\frac{180}{221} + \frac{40}{221}}{\frac{75}{221} + \frac{96}{221}} = \frac{-\frac{140}{221}}{\frac{171}{221}} = -\frac{140}{171}$.

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 $\tan ( \alpha - \beta )$