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Chapter 5: Problem 39

Find the area of the triangle having the indicated angle and sides. $$C=120^{\circ}, \quad a=4, \quad b=6$$

Short Answer

Expert verified
The area of the triangle is approximately 10.39 square units.

Step by step solution

01

Identify the sides and angle

We are given sides \( a = 4 \), \( b = 6 \), and angle \( C = 120^{\circ} \). The area of a triangle when two sides and the included angle are given can be calculated with the formula \( A = \frac{1}{2}ab\sin(C) \)
02

Convert the angle to radians

Before substituting the values into the formula, be sure to convert the measure of angle C from degrees to radians because the trigonometric functions in most calculators are radian-based. We can convert degrees to radians using the relation: \( \text{radian} = \text{degree} 脳 \frac{\pi}{180} \). Substitute \( C = 120^{\circ} \) and calculate.
03

Substitute the values into the formula

Substitute the values of \( a \), \( b \), and \( C \) (in radians) into the formula: \( A = \frac{1}{2} \times 4 \times 6 \times \sin(120 \times \frac{\pi}{180}) \)
04

Compute the area

Perform the multiplication and calculate the sine function to obtain an answer for the area. Remember to round the final result to a suitable number of decimal places.

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