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

step 1 So we know that sine of theta is equal to negative 1 3. And theta is between 270 degrees and 360

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.
$$
\sin \theta=-\frac{1}{3} ; 270^{\circ}<\theta<360^{\circ}
$$

Key Concepts

Quadrant Analysis

Quadrant analysis is the process of determining the sign (positive or negative) of a trigonometric function based on the angle's location in the coordinate plane. This concept is crucial when computing exact values, especially when taking square roots in half-angle formulas or when deciding the appropriate form of double angle formulas.

Pythagorean Identity

The Pythagorean identity, sin²? + cos²? = 1, is fundamental in relating the sine and cosine of an angle. It allows one to solve for the cosine when the sine is known (or vice versa), thereby providing the necessary values to apply other trigonometric identities such as double and half angle formulas.

Double Angle Identities

Double angle identities express trigonometric functions of twice an angle in terms of functions of the original angle. For example, sin 2? can be written as 2 sin ? cos ? and cos 2? as either cos²? ? sin²? or 1 ? 2 sin²?, which provide multiple ways to compute these values when sin ? or cos ? is known.

Half Angle Identities

Half angle identities enable the computation of trigonometric functions of half an angle using the known trigonometric values of the full angle. They are given by formulas such as sin(?/2) = ±?((1 ? cos?)/2) and cos(?/2) = ±?((1 + cos?)/2), where the correct sign is determined by analyzing the quadrant of the resulting angle.

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