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Chapter 6: Problem 43

Describe identities that can be verified using the sum-to-product formulas.

Short Answer

Expert verified
The sum-to-product identities can be used to verify mathematical identities involving the sum or difference of sine or cosine functions. By plugging in certain angles into the identity and checking if both sides of the equation result in the same output, the identity can be verified.

Step by step solution

01

Understanding Sum-to-Product Formulas

Sum-to-product identities involve the rewrite of the sum or difference of sine or cosine functions as a product of sine or cosine functions. The identities are as follows:1. \[\sin x +\sin y = 2\sin\frac{x+y}{2}\cos\frac{x-y}{2}\]2. \[\sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2}\]3. \[\cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2}\]4. \[\cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\]
02

Applying Sum-to-Product Formulas

The sum-to-product formulas can be used to verify identities involving the sum or difference of sine or cosine functions. Suppose we are given the identity \(\sin a + \sin b = 2\sin\frac{a+b}{2}\cos\frac{a-b}{2}\). To verify the identity, replace \(a\) and \(b\) with any angle (preferably in the first quadrant from where the trigonometric values are positive and easy to calculate) and then confirm that both sides of the equation yield the same result.
03

Verifying the Identity

Let's verify the identity for \(a=30^{\circ}\) and \(b=45^{\circ}\). These angles will be just right to have calculable values of sines and cosines. Place these values in above identity:On right hand side: \[ RHS = 2\sin\frac{30^{\circ}+45^{\circ}}{2}\cos\frac{30^{\circ}-45^{\circ}}{2}=2\sin37.5^{\circ}\cos7.5^{\circ}\]On left hand side: \[ LHS = \sin 30^{\circ} + \sin 45^{\circ}\]If both LHS and RHS return the same result, then the identity can be identified as correct.

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Most popular questions from this chapter

solve each equation on the interval \([0,2 \pi) .\) \(2 \cos ^{3} x+\cos ^{2} x-2 \cos x-1=0\) (Hint: Use factoring by grouping.)

Write each trigonometric expression as an algebraic expression (that is, without any trigonometric functions). Assume that x and y are positive and in the domain of the given inverse trigonometric function. $$ \cos \left(\sin ^{-1} x-\cos ^{-1} y\right) $$

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$ \tan x=-6.2154 $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equations \(\sin 2 x=1\) and \(\sin 2 x=\frac{1}{2}\) have the same number of solutions on the interval \([0,2 \pi)\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I solved \(\cos \left(x-\frac{\pi}{3}\right)=-1\) by first applying the formula for the cosine of the difference of two angles.
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