Exploring the Sum-to-Product Identity and Angle Addition Formulas in Trigonometry | How to Find sin(A+B)

sin(A+B)

The sine of the sum of two angles, A and B, denoted as sin(A+B), can be expressed using trigonometric identities

The sine of the sum of two angles, A and B, denoted as sin(A+B), can be expressed using trigonometric identities.

First, let’s define some identities that will be useful in finding sin(A+B).

1. Sum-to-Product Identity:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

2. Angle Addition Formulas:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
cos(A + B) = cos(A)cos(B) – sin(A)sin(B)

Using these identities, we can find sin(A+B) if we know the values of sin(A), sin(B), cos(A), and cos(B).

For example, let’s consider sin(30° + 45°):
If we know sin(30°) = 0.5, sin(45°) = √2 / 2, cos(30°) = √3 / 2, and cos(45°) = √2 / 2, we can substitute these values into the sum-to-product identity:

sin(30° + 45°) = sin(30°)cos(45°) + cos(30°)sin(45°)
= (0.5)(√2 / 2) + (√3 / 2)(√2 / 2)
= √2 / 4 + √6 / 4
= (√2 + √6) / 4

So, sin(30° + 45°) is equal to (√2 + √6) / 4.

By using the given values of sin(A), sin(B), cos(A), and cos(B), you can substitute them into the relevant trigonometric identities and simplify to find the value of sin(A + B).

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