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

Ask by Herrera Simpson. in the United States

Jan 23,2025

Question

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

UpStudy AI · Accepted Answer

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

we can use the sine addition formula:
$$
\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
$$

**Step 1: Find $$\cos \alpha$$**
$$
\sin^2 \alpha + \cos^2 \alpha = 1 \
\left(\frac{7}{25}\right)^2 + \cos^2 \alpha = 1 \
\frac{49}{625} + \cos^2 \alpha = 1 \
\cos^2 \alpha = 1 - \frac{49}{625} = \frac{576}{625} \
\cos \alpha = \frac{24}{25} \quad (\text{since } \alpha \text{ is in the first quadrant})
$$

**Step 2: Find $$\sin \beta$$**
$$
\cos^2 \beta + \sin^2 \beta = 1 \
\left(\frac{8\sqrt{89}}{89}\right)^2 + \sin^2 \beta = 1 \
\frac{64 \times 89}{89^2} + \sin^2 \beta = 1 \
\frac{64}{89} + \sin^2 \beta = 1 \
\sin^2 \beta = 1 - \frac{64}{89} = \frac{25}{89} \
\sin \beta = -\frac{5}{\sqrt{89}} = -\frac{5\sqrt{89}}{89} \quad (\text{since } \beta \text{ is in the fourth quadrant})
$$

**Step 3: Apply the sine addition formula**
$$
\sin(\alpha + \beta) = \left(\frac{7}{25}\right)\left(\frac{8\sqrt{89}}{89}\right) + \left(\frac{24}{25}\right)\left(-\frac{5\sqrt{89}}{89}\right) \
= \frac{56\sqrt{89}}{2225} - \frac{120\sqrt{89}}{2225} \
= \frac{-64\sqrt{89}}{2225}
$$

**Final Answer:**
$$
\sin(\alpha + \beta) = -\frac{64\, \sqrt{89}}{2225}
$$

Answer:
$$\sin(\alpha + \beta) = -\,\dfrac{64\,\sqrt{89}}{2225}$$
$$\sin(\alpha + \beta) = -\frac{64\sqrt{89}}{2225}$$

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

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