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

step 1 We know that cosine of theta is equal to negative 1, 4, and theta is in quadrant 2, because we k

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.
$$
\cos \theta=-\frac{1}{4} ; 90^{\circ}<\theta<180^{\circ}
$$

Key Concepts

Pythagorean Identity

The Pythagorean identity, sin² ? + cos² ? = 1, provides a fundamental relationship between the sine and cosine functions. It is frequently used to determine one trigonometric function when the other is known. This identity underpins the derivation of many trigonometric formulas and is key to solving problems involving trigonometric expressions and transformations.

Quadrant Sign Considerations

In trigonometry, the sign of a trigonometric function is determined by the quadrant in which its angle lies. This knowledge is vital when applying formulas like the half-angle formulas. For instance, based on the specific range provided for an angle (or its half), one must identify whether the resulting sine or cosine value should be positive or negative, ensuring correct evaluation in problems involving angular transformations.

Double Angle Formulas

These formulas allow one to express trigonometric functions of twice an angle in terms of the trigonometric functions of the original angle. For example, sin 2? can be expressed as 2 sin ? cos ? and cos 2? as 2 cos^2 ? - 1 (or equivalently 1 - 2 sin^2 ?). They are essential for finding exact values of trigonometric expressions when the value of a single trigonometric function is known.

Half-Angle Formulas

Half-angle formulas are used to calculate the sine and cosine of half an angle using the cosine of the original angle. They are given by sin(?/2) = ±?((1 - cos ?)/2) and cos(?/2) = ±?((1 + cos ?)/2). The determination of the sign depends on the quadrant in which the half-angle lies, making these formulas crucial when working with precise angle measures.

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