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 5 .13th, where theta is in quadrant 2, which means it'

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.
$$
\sin \theta=\frac{5}{13} ; 90^{\circ}<\theta<180^{\circ}
$$

Key Concepts

Double-Angle Identities

Double-angle identities, such as sin 2? = 2 sin? cos? and cos 2? = cos^2? ? sin^2?, are formulas that express trigonometric functions of double angles in terms of single angles. They are particularly useful when an expression needs to be simplified or when exact values of functions at multiple angles are required.

Half-Angle Identities

Half-angle identities are formulas that express the trigonometric functions of half an angle in terms of the cosine of the full angle, for example, sin(?/2) = ±?((1 ? cos?)/2) and cos(?/2) = ±?((1 + cos?)/2). These identities are essential in problems that require splitting an angle into two equal parts, and proper sign selection is based on the quadrant of the resulting half-angle.

Pythagorean Identity

This fundamental trigonometric relation, sin^2? + cos^2? = 1, is used to find an unknown trigonometric function when another is known. In problems where only one trigonometric value is given, the identity allows for the determination of the other function by accounting for the inherent relationship between sine and cosine.

Quadrant Analysis

Quadrant analysis involves determining the sign (positive or negative) of trigonometric functions based on the angle's location on the unit circle. Knowing the quadrant in which the angle lies is crucial, as it affects whether the values determined by identities (like cosine from sine through the Pythagorean identity) should be taken as positive or negative.

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