Question

If cos = square root of 2/2 and 3 pi/2 < theta < 2 pi, what are the values of sin and tan?

Options:
a. Sin = -1/2, Tan = -1
b. Sin = -1/2, Tan = 1
c. Sin = 1/2, Tan = -1
d. Sin = 1/2, Tan = 1

Answer 1

Given that cos(θ) = √2/2 in the fourth quadrant, we find that sin(θ) = -1/2 and tan(θ) = -1, so the correct option is a. Sin = -1/2, Tan = -1.

Step-by-step explanation:

To determine the values of sin and tan given that cos(θ) = √2/2 and 3π/2 < θ < 2π, we first recognize that θ is in the fourth quadrant where cosine is positive, sine is negative, and tangent is negative.

Given that cos(θ) = √2/2, and based on the Pythagorean identity sin^2(θ) + cos^2(θ) = 1, we can find sin(θ) by solving for sin^2(θ):
sin^2(θ) = 1 - cos^2(θ) = 1 - (√2/2)^2 = 1 - 1/2 = 1/2.
Since θ is in the fourth quadrant, sin(θ) must be negative, giving us sin(θ) = -√(1/2) which simplifies to sin(θ) = -1/2.

Next, to find tan(θ), use the relationship tan(θ) = sin(θ)/cos(θ):
τan(θ) = (-1/2) / (√2/2) = -1.

Therefore, the correct option is a. Sin = -1/2, Tan = -1.