The area of a triangle is $10$ . Two of the side lengths are $5$ . $3$ and $5$ and the included angle is obtuse. Find the measure of the included angle, to the nearest tenth of a degree. $\newline$ Answer:

View step-by-step help

Home

Math Problems

Precalculus

Inverses of trigonometric functions using a calculator

Full solution

Q. The area of a triangle is

10

. Two of the side lengths are

5

3

and

5

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

\newline

Answer:

Area Formula Application: 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.
Substitute Values: We know the area $A = 10$ , side $a = 5.3$ , and side $b = 5$ . We can plug these values into the area formula to solve for $\sin(C)$ .
Calculate Sine: The formula with the given values is $10 = \frac{1}{2} \times 5.3 \times 5 \times \sin(C)$ .
Simplify Fraction: Solving for $\sin(C)$ , we get $\sin(C) = \frac{10}{\frac{1}{2} \times 5.3 \times 5}$ .
Find Angle: Calculating the right side of the equation, we have $\sin(C) = \frac{10}{13.25}$ .
Use Inverse Sine: Simplifying the fraction, we get $\sin(C) = \frac{20}{26.5}$ .
Determine Obtuse Angle: Further simplifying, we get $\sin(C) = \frac{40}{53}$ .
Determine Obtuse Angle: Further simplifying, we get $\sin(C) = \frac{40}{53}$ .Now, we need to find the angle $C$ whose sine is $\frac{40}{53}$ . Since the angle is obtuse, we are looking for an angle between $90^\circ$ and $180^\circ$ .
Determine Obtuse Angle: Further simplifying, we get $\sin(C) = \frac{40}{53}$ .Now, we need to find the angle $C$ whose sine is $\frac{40}{53}$ . Since the angle is obtuse, we are looking for an angle between $90^\circ$ and $180^\circ$ .Using a calculator, we find the inverse sine (arcsin) of $\frac{40}{53}$ . However, since the sine function is positive in both the first and second quadrants, and we are looking for an obtuse angle (which lies in the second quadrant), we must subtract the arcsin value from $180^\circ$ to find the obtuse angle.
Determine Obtuse Angle: Further simplifying, we get $\sin(C) = \frac{40}{53}$ .Now, we need to find the angle $C$ whose sine is $\frac{40}{53}$ . Since the angle is obtuse, we are looking for an angle between $90^\circ$ and $180^\circ$ .Using a calculator, we find the inverse sine (arcsin) of $\frac{40}{53}$ . However, since the sine function is positive in both the first and second quadrants, and we are looking for an obtuse angle (which lies in the second quadrant), we must subtract the arcsin value from $180^\circ$ to find the obtuse angle.The arcsin of $\frac{40}{53}$ is approximately $49.6^\circ$ . To find the obtuse angle, we calculate $180^\circ - 49.6^\circ$ .
Determine Obtuse Angle: Further simplifying, we get $\sin(C) = \frac{40}{53}$ .Now, we need to find the angle $C$ whose sine is $\frac{40}{53}$ . Since the angle is obtuse, we are looking for an angle between $90^\circ$ and $180^\circ$ .Using a calculator, we find the inverse sine (arcsin) of $\frac{40}{53}$ . However, since the sine function is positive in both the first and second quadrants, and we are looking for an obtuse angle (which lies in the second quadrant), we must subtract the arcsin value from $180^\circ$ to find the obtuse angle.The arcsin of $\frac{40}{53}$ is approximately $49.6^\circ$ . To find the obtuse angle, we calculate $180^\circ - 49.6^\circ$ .The obtuse angle $C$ is approximately $C$ $1$ .