I have tried and tried to understand double angle identities but it just won't stick.
if sin x = -0.6 and 180 (degrees) < x < 270 (degrees), find the exact value of sin 2x
sin 2x = 2 sin x cos x
x is in quadrant 3 so cos x is negative
sin 2x = 2(-0.6) cos x
but
cos^2 x = 1 - sin^2 x
cos^2x = 1 - .36 = .64 = 64 * 10^-2
so
cos x = + or - 8*10^-1 = -0.8 because we know it is -
so
sin 2x = 2 (-0.6)(-0.8) = + 0.96
Understanding double angle identities can be tricky, but with some practice and guidance, you can get the hang of it. Let's break it down step by step.
To find the exact value of sin 2x, we will need to use the double angle identity for sine:
sin 2x = 2sin x cos x
First, we need to find the values of sin x and cos x using the given information.
sin x = -0.6
We know that sin x is negative, which can only happen in the second and third quadrants. Since 180 degrees < x < 270 degrees, we are in the third quadrant.
To find cos x, we can use the Pythagorean identity: sin² x + cos² x = 1
Since we know sin x, we can rearrange this equation to solve for cos x:
cos² x = 1 - sin² x
cos² x = 1 - (-0.6)²
cos² x = 1 - 0.36
cos² x = 0.64
Taking the square root of both sides, we get:
cos x = ± √0.64
cos x = ± 0.8
Since we are in the third quadrant (180 degrees < x < 270 degrees), cos x is negative. Therefore:
cos x = -0.8
Now that we have the values of sin x and cos x, we can substitute them into the double angle identity:
sin 2x = 2sin x cos x
sin 2x = 2(-0.6)(-0.8)
sin 2x = 0.96
So, the exact value of sin 2x is 0.96.
Remember, understanding double angle identities takes practice. It's helpful to familiarize yourself with the unit circle and the trigonometric values for different angles.