Question

find the exact value of the expressions \\( \cos ( \alpha + \beta ) \\), \\( \sin ( \alpha + \beta ) \\) and \\( \tan ( \alpha + \beta ) \\) under the following conditions: \\( \sin ( \alpha ) = \frac { 24 } { 25 } \\), \\( \alpha \\) lies in quadrant i, and \\( \sin ( \beta ) = \frac { 4 } { 5 } \\), \\( \beta \\) lies in quadrant ii. a. \\( \cos ( \alpha + \beta ) = \frac { - 117 } { 125 } \\) (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.) b. \\( \sin ( \alpha + \beta ) = - \frac { 44 } { 125 } \\) (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.) c. \\( \tan ( \alpha + \beta ) = \square \\) (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Calculate $\cos\alpha$ and $\cos\beta$

Using the Pythagorean identity $\sin^{2}\theta+\cos^{2}\theta = 1$.
For $\alpha$:
Since $\sin\alpha=\frac{24}{25}$ and $\alpha$ is in quadrant I, then $\cos\alpha=\sqrt{1-\sin^{2}\alpha}=\sqrt{1 - (\frac{24}{25})^{2}}=\sqrt{\frac{625 - 576}{625}}=\sqrt{\frac{49}{625}}=\frac{7}{25}$.
For $\beta$:
Since $\sin\beta=\frac{4}{5}$ and $\beta$ is in quadrant II, then $\cos\beta=-\sqrt{1-\sin^{2}\beta}=-\sqrt{1 - (\frac{4}{5})^{2}}=-\sqrt{\frac{25 - 16}{25}}=-\sqrt{\frac{9}{25}}=-\frac{3}{5}$.

Step2: Calculate $\tan\alpha$ and $\tan\beta$

Using the formula $\tan\theta=\frac{\sin\theta}{\cos\theta}$.
$\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=\frac{\frac{24}{25}}{\frac{7}{25}}=\frac{24}{7}$.
$\tan\beta=\frac{\sin\beta}{\cos\beta}=\frac{\frac{4}{5}}{-\frac{3}{5}}=-\frac{4}{3}$.

Step3: Calculate $\tan(\alpha+\beta)$

Using the formula $\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$.
Substitute $\tan\alpha=\frac{24}{7}$ and $\tan\beta =-\frac{4}{3}$ into the formula:

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Answer:

$\frac{44}{117}$