/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 41 Find the area of the triangle ha... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 41

Find the area of the triangle having the indicated angle and sides. $$A=43^{\circ} 45^{\prime}, \quad b=57, \quad c=85$$

Short Answer

Expert verified
Put the calculations into the calculator as following: \(\frac{1}{2} \times 57 \times 85 \times \sin 43.75\) radians equals approximately 1081.69 square units. So, the area of the triangle is approximately 1081.69 square units.

Step by step solution

01

Conversion of the angle from degrees and minutes to pure degrees

Given the angle A as \(43^{\circ} 45^{\prime}\), it can be converted into pure degrees using the conversion \(1^{\circ} = 60^{\prime}\). So, \(45^{\prime}\) equals \(\frac{45}{60} = 0.75^{\circ}\). Adding that to the original degree portion, the complete angle A in degrees becomes \(43.75^{\circ}\).
02

Substitution of the Values into the Formula

The formula to calculate the area of a triangle when two sides and the included angle are given is \[Area = \frac{1}{2} \times b \times c \times \sin A\]. Substitute the degrees into the formula, remember to convert degrees into radians, as most calculators take radians as input for trigonometric calculations. A radian=degree \times \(\frac{\pi}{180}\). Thus, \(A = 43.75 \times \frac{\pi}{180}\). Now putting the values into the formula, we get \[Area = \frac{1}{2} \times 57 \times 85 \times \sin 43.75\) radians. Now solving the calculation will give the answer.
03

Calculate the Area

Using a scientific calculator, put the value from previous step into the calculator. The result is the area of the triangle.

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