The area of a triangle is 706. Two of the side lengths are 45 and 33 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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The area of a triangle is 706. Two of the side lengths are 45 and 33 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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Calculate Area Formula: To find the measure of the included angle, we can use the formula for the area of a triangle when two sides and the included angle are known: Area = (1/2) * a * b * sin(C), where a and b are the side lengths, and C is the included angle. Given that the area is 706 square units, and the side lengths are 45 and 33 units, we can set up the equation: 706 = (1/2) * 45 * 33 * sin(C)Set Up Equation: Now, we need to solve for sin(C). First, we multiply 45 and 33, and then divide 706 by the result, and finally multiply by 2 to isolate sin(C) on one side of the equation: sin(C) = (2 * 706) / (45 * 33)Solve for sin(C): Perform the calculations to find the value of sin(C): sin(C) = (2 * 706) / (45 * 33) = 1412 / 1485 ≈ 0.951Find Acute Angle: Since the angle is obtuse, it lies in the second quadrant where sine is positive. We need to find the angle whose sine is approximately 0.951. We use the inverse sine function to find the angle, but since the range of arcsin is from -π/2 to π/2, we need to find the supplement of the angle that arcsin would give us because the angle is obtuse. Let's first find the acute angle whose sine is 0.951: C_acute = arcsin(0.951)Calculate Acute Angle: Calculate the acute angle using a calculator: C_acute ≈ arcsin(0.951) ≈ 71.8 degreesFind Obtuse Angle: To find the obtuse angle, we take the supplement of the acute angle: C_obtuse = 180 degrees - C_acuteCalculate Obtuse Angle: Perform the calculation to find the measure of the obtuse angle: C_obtuse = 180 degrees - 71.8 degrees ≈ 108.2 degrees

The area of a triangle is $706$ . Two of the side lengths are $45$ and $33$ 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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The area of a triangle is $706$ . Two of the side lengths are $45$ and $33$ and the included angle is obtuse. Find the measure of the included angle, to the nearest tenth of $a$ degree. $\newline$ Answer: