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Chapter 19: Problem 1833

If \(\sin x \cos y=(1 / 8)\) and \(2 \cot x=3 \cot y\) then \(\sin (x+y)=\) (a) \((1 / 16)\) (b) \((5 / 16)\) (c) \((1 / 8)\) (d) \((5 / 8)\)

Short Answer

Expert verified
The short answer to the given question is: \(\sin (x+y) = \frac{5}{16}\) (b).

Step by step solution

01

Express cotangent in terms of sine and cosine

Recall that \(\cot x = \frac{\cos x}{\sin x}\) and \(\cot y = \frac{\cos y}{\sin y}\). Substitute these expressions into the given equation \(2\cot x = 3\cot y\), we get: \(2 \frac{\cos x}{\sin x} = 3 \frac{\cos y}{\sin y}\)
02

Rearrange the equation

To get an expression for \(\cos x\sin y\), we'll clear the denominators in the equation above by multiplying both sides by \(\sin x \sin y\): \(\Rightarrow 2 (\cos x) (\sin x)(\sin y)=3 (\sin x) (\cos y)(\sin y)\) We're given that \(\sin x \cos y = \frac{1}{8}\). We can rearrange the equation we just derived to isolate \(\cos x\sin y\): \(\Rightarrow \cos x\sin y = \frac{3}{2}(\sin x\cos y)\) Now substitute \(\sin x \cos y = \frac{1}{8}\) into the expression: \(\Rightarrow \cos x\sin y = \frac{3}{2}\times\frac{1}{8} = \frac{3}{16}\)
03

Use the angle addition formula for sine

Now, we're ready to find the value of \(\sin (x+y)\). Recall the angle addition formula for sine: \(\sin (x+y) = \sin x \cos y + \cos x \sin y\) From the given equations, we know that \(\sin x \cos y = \frac{1}{8}\), and we have found that \(\cos x\sin y = \frac{3}{16}\). Substitute these values into the formula: \(\sin (x+y) = \frac{1}{8} + \frac{3}{16}\)
04

Compute the final result

Add the fractions to get the result: \(\sin (x+y) = \frac{1}{8} + \frac{3}{16} = \frac{2}{16} + \frac{3}{16} = \frac{5}{16}\) So, the correct answer is: \(\sin (x+y) = \frac{5}{16}\) (b).

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