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

Using the double-angle formula for sine, we have:

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

Letting x = 11eb80b1_afd6_5cfa_a291_5f7fdef1a29a_TB8633_11, we can find sin(2x):

sin(2x) = 2sin(x)cos(x)
= 2(sin(x))^2   (since cos(x) = sqrt(1 - (sin(x))^2))
= 2(11eb80b1_afd6_5cfa_a291_5f7fdef1a29a_TB8633_11)^2

Now we can use the double-angle formula for cosine to find cos(2x):

cos(2x) = cos^2(x) - sin^2(x)
= 1 - 2(sin(x))^2
= 1 - 2(11eb80b1_afd6_5cfa_a291_5f7fdef1a29a_TB8633_11)^2

Finally, we can substitute these values into the expression 1 + sin(2x)cos(2x) to get:

1 + sin(2x)cos(2x) = 1 + 2(sin(x))^2(1 - 2(sin(x))^2)
= -4(11eb80b1_afd6_5cfa_a291_5f7fdef1a29a_TB8633_11)^4 + 2(11eb80b1_afd6_5cfa_a291_5f7fdef1a29a_TB8633_11)^2 + 1
= 11eb80b1_afd6_ab1f_a291_8135e6236873_TB8633_11

Therefore, the answer is D.

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