Find the exact value of each of the following under the given conditions.

Ask by Chandler Tran. in the United States

Jan 23,2025

Question

Find the exact value of each of the following under the given conditions. $$ \sin \alpha=\frac{7}{25}, 0<\alpha<\frac{\pi}{2} ; \cos \beta=\frac{8 \sqrt{89}}{89},-\frac{\pi}{2}<\beta<0 $$ $$ \begin{array}{llll}	ext { (a) } \sin (\alpha+\beta) & 	ext { (b) } \cos (\alpha+\beta) & 	ext { (c) } \sin (\alpha-\beta) & 	ext { (d) } 	an (\alpha-\beta) \ 	ext { (a) } \sin (\alpha+\beta)=-\frac{64 \sqrt{89}}{2225} \ 	ext { (Type an exact answer using radicals as needed. Rationalize all denominators. Use integers or fractions for any numbers in the expression. Simplify your answer.) } \ 	ext { (b) } \cos (\alpha+\beta)=\frac{227 \sqrt{89}}{2225} & 	ext { (Type an exact answer using radicals as needed. Rationalize all denominators. Use integers or fractions for any numbers in the expression. Simplify your answer.) } \ 	ext { (c) } \sin (\alpha-\beta)=\square & 	ext { (Type an exact answer using radicals as needed. Rationalize all denominators. Use integers or fractions for any numbers in the expression. Simplify your answer.) }\end{array} $$

UpStudy AI · Accepted Answer

To find $$ \sin(\alpha - \beta) $$ given:
- $$ \sin \alpha = \frac{7}{25} $$ with $$ 0 < \alpha < \frac{\pi}{2} $$,
- $$ \cos \beta = \frac{8\sqrt{89}}{89} $$ with $$ -\frac{\pi}{2} < \beta < 0 $$,

follow these steps:

1. **Find $$ \cos \alpha $$:**
   $$
   \cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - \left(\frac{7}{25}\right)^2} = \sqrt{1 - \frac{49}{625}} = \sqrt{\frac{576}{625}} = \frac{24}{25}
   $$

2. **Find $$ \sin \beta $$:**
   Since $$ \beta $$ is in the fourth quadrant, $$ \sin \beta $$ is negative.
   $$
   \sin \beta = -\sqrt{1 - \cos^2 \beta} = -\sqrt{1 - \left(\frac{8\sqrt{89}}{89}\right)^2} = -\sqrt{1 - \frac{64 \cdot 89}{7921}} = -\sqrt{\frac{2225}{7921}} = -\frac{5\sqrt{89}}{89}
   $$

3. **Use the sine difference identity:**
   $$
   \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta
   $$
   Substitute the known values:
   $$
   \sin(\alpha - \beta) = \left(\frac{7}{25}\right)\left(\frac{8\sqrt{89}}{89}\right) - \left(\frac{24}{25}\right)\left(-\frac{5\sqrt{89}}{89}\right)
   $$
   $$
   \sin(\alpha - \beta) = \frac{56\sqrt{89}}{2225} + \frac{120\sqrt{89}}{2225} = \frac{176\sqrt{89}}{2225}
   $$

**Final Answer:**
$$
\sin(\alpha - \beta) = \frac{176\, \sqrt{89}}{2225}
$$
$$ \sin(\alpha - \beta) = \frac{176\, \sqrt{89}}{2225} $$

Find the exact value of each of the following under the given conditions.

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