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Chapter 5: Problem 44

Find the area of the triangle having the indicated angle and sides. $$C=84^{\circ} 30^{\prime}, \quad a=16, \quad b=20$$

Short Answer

Expert verified
The area of the triangle is approximately 144.2 square units.

Step by step solution

01

Convert the Angle to Decimal Degrees

Given angle \(C=84^{\circ} 30^{\prime}\), we need to convert the minutes part into decimal form. This can be done by taking the minutes and dividing it by 60 to get the decimal equivalent. Thus, \(C = 84.5^{\circ}\).
02

Convert the Angle to Radians

Most mathematical functions use angles in radians not in degrees. Therefore, before using the angle in any formula, we need to convert it to radians first. The conversion can be done by using the formula \(\dfrac{\pi radians}{180^{\circ}}\). Thus, \(C = 84.5^{\circ} * \frac{\pi rad}{180^{\circ}} = 1.474 rad\).
03

Apply the Area Formula

Now, we can apply the formula for the area of a triangle with known sides and included angle: \(Area = 0.5*a*b*\sin(C)\). Substituting the given values, we have \(Area = 0.5*16*20*\sin(1.474 rad)\).
04

Compute the Area

Performing the calculation gives the area of the triangle as \(Area \approx 144.2\) square units.

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