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The area of a triangle is 866 . Two of the side lengths are 33 and 98 and the included angle is acute. Find the measure of the included angle, to the nearest tenth of a degree. Answer:

The area of a triangle is 866866 . Two of the side lengths are 3333 and 9898 and the included angle is acute. Find the measure of the included angle, to the nearest tenth of a degree.\newlineAnswer:

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Q. The area of a triangle is 866866 . Two of the side lengths are 3333 and 9898 and the included angle is acute. Find the measure of the included angle, to the nearest tenth of a degree.\newlineAnswer:
  1. Area Calculation: We know the formula for the area of a triangle when two sides and the included angle are given: Area=12⋅a⋅b⋅sin⁡(C)\text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin(C), where aa and bb are the sides and CC is the included angle. We can use this formula to find the measure of the included angle.\newlineArea=12⋅33⋅98⋅sin⁡(C)=866\text{Area} = \frac{1}{2} \cdot 33 \cdot 98 \cdot \sin(C) = 866.
  2. Isolating sin(C): First, we need to isolate sin⁡(C)\sin(C) in the equation.sin⁡(C)=2×Areaa×b=2×86633×98\sin(C) = \frac{2 \times \text{Area}}{a \times b} = \frac{2 \times 866}{33 \times 98}.
  3. Calculating sin⁡(C)\sin(C): Now, we calculate the value of sin⁡(C)\sin(C).sin⁡(C)=17323234≈0.5355\sin(C) = \frac{1732}{3234} \approx 0.5355.
  4. Finding Angle C: To find the angle CC, we need to take the inverse sine (arcsin) of extsin(C) ext{sin}(C). \newlineC=extarcsin(0.5355)C = ext{arcsin}(0.5355).
  5. Calculating Angle CC: Using a calculator, we find the value of CC to the nearest tenth of a degree.\newlineC≈32.5∘C \approx 32.5^\circ.

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