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

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Recall the double-angle identity for cosine: \(\cos 2\theta = 2\cos^2 \theta - 1\).

Substitute the given value \(\cos 2\theta = \frac{1}{2}\) into the identity: \(\frac{1}{2} = 2\cos^2 \theta - 1\).

Solve the equation for \(\cos^2 \theta\): add 1 to both sides to get \(\frac{3}{2} = 2\cos^2 \theta\), then divide both sides by 2 to find \(\cos^2 \theta = \frac{3}{4}\).

Take the square root of both sides to find \(\cos \theta = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}\).

Determine the correct sign of \(\cos \theta\) based on the quadrant: since \(\theta\) terminates in quadrant II, where cosine values are negative, choose \(\cos \theta = -\frac{\sqrt{3}}{2}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Double-Angle Identity for Cosine

The double-angle identity relates cos 2θ to cos θ and sin θ, commonly expressed as cos 2θ = 2cos²θ - 1. This identity allows us to find cos θ when cos 2θ is known by rearranging and solving a quadratic equation.

Quadrant Sign Rules

The sign of trigonometric functions depends on the quadrant of the angle. In quadrant II, cosine values are negative while sine values are positive. This information helps determine the correct sign of cos θ after solving the equation.

Solving Quadratic Equations in Trigonometry

When using identities like the double-angle formula, you often get quadratic equations in terms of cos θ or sin θ. Solving these equations and applying domain restrictions or quadrant information is essential to find the correct trigonometric value.

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