The expression (sin a + sin b) is equivalent to:
A: 2 cos [(a + b) / 2] cos [(a - b) / 2]
B: none of the other answers is correct
C: 2 sin a⋅ cos b
D: 2 sin [(a + b) / 2] sin [(a - b) / 2]
E: 2 sin [(a + b) / 2] cos [(a - b) / 2]
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When working with trigonometry, it’s important to know some useful identities that can help simplify expressions. One such identity is the sum-to-product identity, which states that the sum of two sine waves can be expressed as two new sine waves with different frequencies and amplitudes.
The sum-to-product identity is written as:
sin a + sin b = 2 sin [(a + b) / 2] sin [(a - b) / 2]
Let’s break this down step by step.
First, we have the two sine waves: sin a and sin b. These are simply the sine values of two different angles, a and b.
Next, we want to express the sum of these two sine waves as a product of two new sine waves. To do this, we use the identity on the right-hand side of the equation.
The first part of the right-hand side is 2 sin [(a + b) / 2]. This is simply the amplitude of the new sine wave, which is double the amplitude of the original sine waves (since sin a and sin b have an amplitude of 1). The value inside the sine function, [(a + b) / 2], is the average of the two original angles, a and b.
The second part of the right-hand side is sin [(a - b) / 2]. This is the second new sine wave, with the same amplitude as the first. The value inside the sine function, [(a - b) / 2], is the difference between the two original angles, a and b.
So, when we multiply these two new sine waves together, we get the sum of the original sine waves, sin a + sin b.
If you have any more questions, feel free to ask!
Very impressive, thank you sir
