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Chapter 19: Problem 1818

If \([(\cos A) / 3]=[(\cos B) / 4]=(1 / 5),-(\pi / 2)

Short Answer

Expert verified
The correct answer is (d) \(-6\).

Step by step solution

01

Use the given cosine equations

We are given that \(\frac{\cos A}{3} = \frac{\cos B}{4} = \frac{1}{5}\). First we need to find the values of cosine for A and B using these equations. From the given relationship, we have: \(\cos A = 3 \cdot \frac{1}{5}\) and \(\cos B = 4 \cdot \frac{1}{5}\) Simplifying, we get: \(\cos A = \frac{3}{5}\) and \(\cos B = \frac{4}{5}\)
02

Use the Pythagorean trigonometric identity

In order to find the sine values for angles A and B, we can use the Pythagorean trigonometric identity: \(\sin^2 x + \cos^2 x = 1\). Applying this identity for angles A and B, we get: \[\sin^2 A + \left(\frac{3}{5}\right)^2 = 1\] and \[\sin^2 B + \left(\frac{4}{5}\right)^2 = 1\]
03

Solve for sine values of A and B

From the equations in Step 2, we can find the sine values for A and B: For A: \(\sin^2 A = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25}\) Since \(-\frac{\pi}{2} < A < 0\), A is a fourth quadrant angle, and its sine value will be negative. Thus, \(\sin A = -\frac{4}{5}\) For B: \(\sin^2 B = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25}\) Since \(-\frac{\pi}{2} < B < 0\), B is a fourth quadrant angle, and its sine value will be negative. Thus, \(\sin B = -\frac{3}{5}\)
04

Calculate the value of 3 sin A + 6 sin B

Now we can substitute the found sine values into the expression "3 sin A + 6 sin B": \(3 \sin A + 6 \sin B = 3 \cdot \left(-\frac{4}{5}\right) + 6 \cdot \left(-\frac{3}{5}\right) = -\frac{12}{5} - \frac{18}{5}\) Calculating the sum, we get: \(-\frac{12}{5} - \frac{18}{5} = -\frac{30}{5} = -6\) So, the value of \(3 \sin A + 6 \sin B\) is \(-6\). Therefore, the correct answer is (d) \(-6\).

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