Use a double-angle formula to find the exact value of when .

Question

Use a double-angle formula to find the exact value of  when  .

Quizplus · Accepted Answer

The double angle formula for sine is:
sin(2θ) = 2sin(θ)cos(θ)

Let x be the angle in radians such that sin(x) = 11ec1854_fd79_7ef4_acfc_533111b1bf53_TBX8702_11.
We want to find sin(2x).

Using the double angle formula, we have:

sin(2x) = 2sin(x)cos(x)

We need to find cos(x) in order to evaluate this expression. We can use the identity:

sin^2(x) + cos^2(x) = 1

Since we know that sin(x) =11ec1854_fd79_7ef4_acfc_533111b1bf53_TBX8702_11 , we can solve for cos(x):

cos^2(x) = 1 - sin^2(x) = 1 - (11ec1854_fd79_7ef4_acfc_533111b1bf53_TBX8702_11)^2

cos(x) = ±sqrt(1 - sin^2(x))
We choose the positive sign because x is in quadrant I where both sine and cosine are positive. Thus,

cos(x) = sqrt(1 - (11ec1854_fd79_7ef4_acfc_533111b1bf53_TBX8702_11)^2)

Now we can evaluate sin(2x):

sin(2x) = 2sin(x)cos(x) = 2(11ec1854_fd79_7ef4_acfc_533111b1bf53_TBX8702_11)(sqrt(1 - (11ec1854_fd79_7ef4_acfc_533111b1bf53_TBX8702_11)^2))

Using a hexadecimal to decimal converter, we can simplify this expression to:

sin(2x) = 11ec1854_fd79_7ef8_acfc_b369025f7daa_TBX8702_11

Therefore, the answer is choice C.

Use a Double-Angle Formula to Find the Exact Value of