Use the given information to find each of the following.

Question

cos θ, given cos 2θ = 1/2 and θ terminates in quadrant II

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the given trigonometric expression using the half angle identity. We are provided that the cosine of X is what we're looking for. If the cosine of two X equals 3/4 and X terminates in quadrant two. Our answer choices are a, the square root of 14 divided by four B negative square root of 14 divided by four C negative radical seven divided by four D radical seven divided by four. To begin with, I want to use half angle identities. So as much as we know, it's the cosine of two X and your brain automatically thinks there's a double angle identity. We wanna use a half angle identity. So recall the half angle identity for cosine is the cosine of an angle A divided by two equals plus or minus the square root of one plus the cosine of A, all of which is divided by two. So here two X is R A because we want to find X. So two X is R A, we can say that we are looking for the cosine of X when this equals plus or minus the square root of one plus. So it's the cosine of two X, which we are told is 3/4 all of which is divided by two. So the cosine of X will equal plus or minus the square root of seven fourths. When I combine one plus 3/4 divided by two and seven fourths divided by two, we will get plus or minus the square root of 7/8 as our cosine of X. So we are in quadrant two. So cosine will be negative. So we're going to have the cosine of X equals negative. And then we have the square root of seven divided by the square root of eight, the square root of eight can be rewritten. So we'll have negative square root of seven divided by two radical two. Rationalizing my denominator, I'm multiplying numerator and denominator by the square to two. So we will get negative radical 14 divided by four. And that's gonna be our value for the cosine of X. And that is answer choice B have a nice day.

Use the given information to find each of the following.
cos θ, given cos 2θ = 1/2 and θ terminates in quadrant II

Key Concepts

Double-Angle Identity for Cosine

Quadrant Sign Rules

Solving Quadratic Equations in Trigonometry

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