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The area of a triangle is 381 . Two of the side lengths are 10 and 93 and the included angle is acute. Find the measure of the included angle, to the nearest tenth of a degree.
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The area of a triangle is $381$ . Two of the side lengths are $10$ and $93$ and the included angle is acute. Find the measure of the included angle, to the nearest tenth of a degree. $\newline$ Answer:

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Find trigonometric ratios using a Pythagorean or reciprocal identity

Full solution

Q. The area of a triangle is

381

. Two of the side lengths are

10

and

93

and the included angle is acute. Find the measure of the included angle, to the nearest tenth of a degree.

\newline

Answer:

Area Formula Explanation: The area of a triangle can be calculated using the formula: $\newline$ Area = $(1/2) \times a \times b \times \sin(C)$ $\newline$ where $a$ and $b$ are the lengths of two sides, and $C$ is the included angle between those sides. We are given that the area is $381$ , and the side lengths are $10$ and $93$ . Let's denote the included angle as $\theta$ .
Rearranging Formula: We can rearrange the formula to solve for $\sin(\theta)$ : $\sin(\theta) = \frac{2 \times \text{Area}}{a \times b}$ Plugging in the given values, we get: $\sin(\theta) = \frac{2 \times 381}{10 \times 93}$
Calculate $\sin(\theta):$ Now, let's calculate the value: $\newline$ $\sin(\theta) = \frac{762}{930}$ $\newline$ $\sin(\theta) = 0.8193548387$
Find Angle $\theta$ : To find the angle $\theta$ , we need to take the inverse sine (arcsin) of the value we calculated: $\newline$ $\theta = \arcsin(0.8193548387)$
Calculate Angle $\theta$ : Using a calculator to find the value of $\theta$ to the nearest tenth of a degree, we get: $\newline$ $\theta \approx 55.1^\circ$

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