Problem 43 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: 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. \[\cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\]

Step 2: 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.

Step 3: 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.