/*! 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 45 Find a nice formula for \(\sin (... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 45

Find a nice formula for \(\sin (5 \theta)\) in terms of \(\sin \theta\).

Short Answer

Expert verified
The nice formula for \(\sin(5\theta)\) in terms of \(\sin \theta\) is given by: \(\sin(5\theta) = 16\sin^5 \theta - 20\sin^3 \theta + 5\sin \theta\)

Step by step solution

01

Write 5胃 as a sum of angles

First, we need to express \(5\theta\) as a sum of angles that we can later use the sum-to-product formulas on. We will express \(5\theta\) as the sum of two angles, say \(m\theta\) and \(n\theta\), with both \(m\) and \(n\) being integers. One way to do this is by writing \(5\theta = 2\theta + 3\theta\).
02

Apply the sum-to-product formula for sine

Now that we have \(5\theta = 2\theta + 3\theta\), we can apply the sum-to-product formula for sine: \(\sin (A + B) = \sin A \cos B + \cos A \sin B\) Substituting \(A = 2\theta\) and \(B = 3\theta\) into the formula: \(\sin (5\theta) = \sin (2\theta + 3\theta) = \sin (2\theta) \cos (3\theta) + \cos (2\theta) \sin(3\theta)\)
03

Apply double and triple angle formulas

Now, let's apply the double and triple angle formulas for sine and cosine to simplify the obtained expression: Double Angle Formulas: \(\sin(2\theta) = 2\sin\theta \cos\theta\) \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\) Triple Angle Formulas: \(\sin(3\theta) = 3\sin\theta - 4\sin^3\theta\) \(\cos(3\theta) = 4\cos^3\theta-3\cos\theta\)
04

Substitute the double and triple angle formulas

Now replace the double and triple angle formulas into the expression we derived in step 2: \(\sin(5\theta) = (2\sin\theta \cos\theta) (4\cos^3\theta-3\cos\theta) + (\cos^2\theta - \sin^2\theta) (3\sin\theta - 4\sin^3\theta)\)
05

Simplify the expression

Now, expand the product terms and simplify the expression: \(\sin(5\theta) = 8\sin\theta\cos^4\theta - 6\sin\theta\cos^2\theta - 2\sin\theta \cos\theta + 3\sin\theta \cos^2\theta - 4\sin^3\theta \cos^2\theta\) We can factor out \(\sin\theta\) from all terms: \(\sin(5\theta) = \sin\theta(8\cos^4\theta - 5\cos^2\theta - 4\sin^2\theta\cos^2\theta + 2\cos\theta)\) Now substitute the identity \(\cos^2\theta = 1 - \sin^2\theta\): \(\sin(5\theta) = \sin\theta(8(1-\sin^2\theta)^2-5(1-\sin^2\theta)-4\sin^2\theta(1-\sin^2\theta)+2(1-\sin^2\theta))\)
06

Finalize the formula

Now, expand and simplify the expression to obtain the final formula for \(\sin(5\theta)\) in terms of \(\sin \theta\): \(\sin(5\theta) = 16\sin^5 \theta - 20\sin^3 \theta + 5\sin \theta\) We now have a nice formula for \(\sin(5\theta)\) in terms of \(\sin\theta\).

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