Question

If sin(theta) = 2/5 and cos(theta) < 0, find 2θ.

a) 2sin⁻¹2/5
b) 2cos⁻¹2/5
c) 2tan⁻¹2/5
d) 2csc⁻¹2/5

Answer 1

To find 2θ, we can use the given values of sin(θ) and cos(θ) to calculate the value of sin(2θ).

Step-by-step explanation:

To find 2θ, we need to first determine the value of θ using the given information. We are given that sin(θ) = 2/5 and cos(θ) < 0. Since sin(θ) = 2/5, we can use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 to find cos(θ). Using this identity, we have (2/5)^2 + cos^2(θ) = 1. Solving for cos(θ), we find that cos(θ) = -4/5.

Now that we know the values of sin(θ) and cos(θ), we can use the double-angle identity for sine to find sin(2θ): sin(2θ) = 2sin(θ)cos(θ). Substituting the values we found earlier, we have sin(2θ) = 2*(2/5)*(-4/5) = -16/25.

In conclusion, the value of 2θ is -16/25.