Question

Given that sin(��) = -7/25 and pi <�� < 3pi/2, what is the exact solution of sin(2��)?

1) 49/625
2) 336/625
3) 527/625
4) 576/625

Answer 1

To find the exact solution of sin(2θ), we can use the double-angle formula for sine. Given that sin(θ) = -7/25 and π < θ < 3π/2, the exact solution of sin(2θ) is -336/625.

Step-by-step explanation:

To find the exact solution of sin(2θ), we can use the double-angle formula for sine.

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

Given that sin(θ) = -7/25 and π < θ < 3π/2, we can substitute these values into the formula.

sin(2θ) = 2(-7/25)cos(θ)

Now, we need to find the value of cos(θ). Since sin(θ) = -7/25 and we know that π < θ < 3π/2, we are in the third quadrant where cosine is negative. Using the Pythagorean identity sin^2(θ) + cos^2(θ) = 1, we can find cos(θ) as follows:

cos(θ) = sqrt(1 - sin^2(θ)) = sqrt(1 - (-7/25)^2) = sqrt(1 - 49/625) = sqrt(576/625) = 24/25

Substituting back into the formula for sin(2θ), we have:

sin(2θ) = 2(-7/25)(24/25) = -336/625

Therefore, the exact solution of sin(2θ) is -336/625.