The area of a triangle is $6$ . Two of the side lengths are $9$ . $6$ and $1$ . $5$ 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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Q. The area of a triangle is

6

. Two of the side lengths are

9

6

and

1

5

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

\newline

Answer:

Set up equation: 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: $\text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin(C)$ , where $a$ and $b$ are the sides and $C$ is the included angle. $\newline$ Given that the area is $6$ , and the sides are $9.6$ and $1.5$ , we can set up the equation: $\newline$ $6 = \frac{1}{2} \cdot 9.6 \cdot 1.5 \cdot \sin(C)$
Solve for sin(C): Now we need to solve for sin(C). First, we multiply $9.6$ and $1.5$ , then divide both sides of the equation by the result to isolate sin(C): $\sin(C) = \frac{2 \times 6}{9.6 \times 1.5}$
Calculate sin(C): Perform the calculations to find the value of sin(C): $\newline$ $\sin(C) = \frac{12}{14.4}$ $\newline$ $\sin(C) = 0.8333\ldots$
Find angle C: Since the angle is obtuse, it is between $90$ degrees and $180$ degrees. The sine function is positive in both the first and second quadrants, but we are looking for an angle in the second quadrant (since it's obtuse). $\newline$ To find the angle $C$ , we take the inverse sine (arcsin) of the value we found for $ext{sin}(C)$ . However, since arcsin will give us an angle less than $90$ degrees, we need to subtract that angle from $180$ degrees to find the obtuse angle. $\newline$ Let's first find the acute angle: $\newline$ $C = ext{arcsin}(0.8333...)$
Find acute angle: Using a calculator to find the $\arcsin(0.8333\ldots)$ gives us an acute angle. However, we need to ensure that the calculator is set to degree mode, not radian mode, to get the correct angle measurement. $\newline$ $C_{\text{acute}} \approx \arcsin(0.8333\ldots) \approx 56.4$ degrees
Find obtuse angle: Now we subtract the acute angle from $180$ degrees to find the obtuse angle: $\newline$ $C_{\text{obtuse}} = 180 - C_{\text{acute}}$ $\newline$ $C_{\text{obtuse}} = 180 - 56.4$ $\newline$ $C_{\text{obtuse}} = 123.6$ degrees