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Chapter 5: Problem 28

Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$A=76^{\circ}, \quad a=34, \quad b=21$$

Short Answer

Expert verified
The first solution would be \(B = 27.32^\circ, C = 76.68^\circ, c = 37.56\) and the second solution would be \(B = 152.68^\circ, C = -49.32^\circ, c = -8.43\). Negative values aren't possible in the context of measurements of angles and lengths in a triangle, thus only one solution exists.

Step by step solution

01

Calculate the height of the triangle

The height (h) of the triangle can be found using the formula \(h = b \cdot \sin(A)\). So plug in the known values: \(h = 21 \cdot \sin(76)\). This gives a rounded value of h = 20.34.
02

Check for the number of possible solutions

Since the length of the opposite side 'a' is longer than h (a > h), there will be two solutions for angle B.
03

Calculate angle B

Use the law of sines to obtain angle B and its supplement if the value is less than 180. First find B as \(\sin^{-1}(\frac{a \cdot \sin(A)}{b})\), then find its supplement as \(180 - B\). Substitute the known values: \(\sin^{-1}(\frac{34 \cdot \sin(76)}{21})\) to get B.
04

Find angle C

Once angles A and B are known, angle C can be found by subtracting the sum of angles A and B from 180 for both possibilities of angle B: \(C = 180 - A - B\).
05

Finding side c

Use the law of sines to find side c for both possibilities. The formula used is \(c = \frac{b \cdot \sin(C)}{\sin(B)}\). Substitute the values to find c.

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Most popular questions from this chapter

Solve the multiple-angle equation. $$\cos \frac{x}{2}=\frac{\sqrt{2}}{2}$$

Write the expression as the sine, cosine, or tangent of an angle. $$w\sin 3 \cos 1.2-\cos 3 \sin 1.2$$

Find the exact value of each expression. (a) \(\cos \left(\frac{\pi}{4}+\frac{\pi}{3}\right)\) (b) \(\cos \frac{\pi}{4}+\cos \frac{\pi}{3}\)

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{13 \pi}{12}$$

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{17 \pi}{12}=\frac{9 \pi}{4}-\frac{5 \pi}{6}$$
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