Find the exact values of sin2 θ, cos2 θ, sin(θ)/(2), and cos(θ)/(2) for each of the following. c

step 1 This question we're given that cosine of theta is equal to negative 2 divided by 3, where theta

Find the exact values of sin2 θ, cos2 θ, sin(θ)/(2), and...

Find the exact values of $\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2},$ and $\cos \frac{\theta}{2}$ for each of the following.
$$
\cos \theta=-\frac{2}{3} ; 180^{\circ}<\theta<270^{\circ}
$$

Key Concepts

Double-Angle Identities

Double-angle identities are formulas in trigonometry that express trigonometric functions of twice an angle (2?) in terms of functions of the original angle (?). These identities, such as sin(2?) = 2 sin ? cos ? and cos(2?) = cos²? ? sin²? (or its variants), are useful for simplifying expressions and computing exact values when given information about sin ? or cos ?.

Half-Angle Identities

Half-angle identities allow the expression of trigonometric functions of half an angle (?/2) in terms of functions of the full angle (?). They take forms like sin(?/2) = ±?((1 ? cos ?)/2) and cos(?/2) = ±?((1 + cos ?)/2). The choice of the positive or negative square root depends on the quadrant in which the resulting angle lies.

Pythagorean Identity

The Pythagorean identity, expressed as sin²? + cos²? = 1, is a fundamental relationship in trigonometry that ties together the sine and cosine functions. It is particularly useful for determining one function when the other is known, enabling the calculation of exact trigonometric values in problems involving angles.

Quadrant Analysis

Quadrant analysis is the method of determining the sign (positive or negative) of trigonometric functions based on the location of the angle on the unit circle. Since the trigonometric functions change their signs depending on the quadrant in which the angle lies, careful analysis of these quadrants is crucial when applying half-angle and double-angle formulas to ensure the correctness of the sign in the computed result.

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