The area of a triangle is 4. Two of the side lengths are 7.5 and 1.6 and the included angle is obtuse. Find the measure of the included angle, to the nearest tenth of a degree. Answer:

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Area Formula: To find the measure of the included angle, we can use the formula for the area of a triangle, which is given by $ A = \frac{1}{2}ab\sin(C) $, where $ A $ is the area, $ a $ and $ b $ are the lengths of two sides, and $ C $ is the included angle between those sides. Given:  Area $ A = 4 $ square units, Side $ a = 7.5 $, Side $ b = 1.6 $. We need to solve for $ C $.Calculate Sin(C): First, we rearrange the area formula to solve for $ \sin(C) $: $ \sin(C) = \frac{2A}{ab} $. Substitute the given values into the formula: $ \sin(C) = \frac{2 \times 4}{7.5 \times 1.6} $.Find Acute Angle: Now, we calculate the value of $ \sin(C) $: $ \sin(C) = \frac{8}{12} $. $ \sin(C) = \frac{2}{3} $.Calculate Obtuse Angle: Since the angle $ C $ is obtuse, we need to find the angle whose sine is $ \frac{2}{3} $ and is greater than 90 degrees but less than 180 degrees. We use the inverse sine function, also known as arcsin, to find the angle. However, since the range of arcsin is typically from -90 to 90 degrees, we need to use the fact that $ \sin(180^\circ - x) = \sin(x) $ to find the obtuse angle.Subtract to Find Angle: We first find the acute angle whose sine is $ \frac{2}{3} $: $ x = \arcsin\left(\frac{2}{3}\right) $. We calculate this value using a calculator.The calculator gives us the acute angle $ x $ which is approximately 41.8 degrees. Since we are looking for the obtuse angle, we calculate $ 180^\circ - x $ to find the measure of the included obtuse angle $ C $: $ C = 180^\circ - 41.8^\circ $.Now, we perform the subtraction to find the measure of the obtuse angle: $ C = 138.2^\circ $. We round this to the nearest tenth of a degree as requested.

The area of a triangle is $4$ . Two of the side lengths are $7$ . $5$ and $1$ . $6$ and the included angle is obtuse. Find the measure of the included angle, to the nearest tenth of a degree. $\newline$ Answer:

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