Solve the equation for exact solutions in the interval 0 ≤ x < 2π. (Enter your answers as a comma-separated list.)
sin 2x cos x + cos 2x sin x = 0
What is the identity for Sin(A+B)?
Ahhh, good observation bob
I did a similar type the long way few posts back,
could have done it this way as well, would have been shorter
To solve the equation sin(2x)cos(x) + cos(2x)sin(x) = 0, we can use trigonometric identities to simplify the expression.
First, let's rewrite the equation as:
sin(2x)cos(x) - sin(x)cos(2x) = 0
Now, we can use the identity sin(A-B) = sin(A)cos(B) - cos(A)sin(B). Applying this identity, we get:
sin(2x - x) = 0
Simplifying further, we have:
sin(x) = 0
The solutions for sin(x) = 0 in the interval 0 ≤ x < 2π are x = 0 and x = π. These are known as the principal values of sin(x) = 0 in the interval.
Therefore, the exact solutions to the equation sin(2x)cos(x) + cos(2x)sin(x) = 0 in the interval 0 ≤ x < 2π are x = 0 and x = π.