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

step 1 So we know that cosine of Theta is equal to negative 1 third, and a theta is in the third quadra

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.
$$
\cos \theta=-\frac{1}{3} ; 180^{\circ}<\theta<270^{\circ}
$$

Key Concepts

Double-Angle Formulas

Double-angle formulas provide expressions for trigonometric functions of twice an angle in terms of functions of the original angle. For example, sin(2?) can be expressed as 2 sin? cos?, and cos(2?) as 2 cos²? - 1 (among other forms). These identities are essential in transforming and simplifying trigonometric expressions where the angle is doubled.

Half-Angle Formulas

Half-angle formulas allow computation of trigonometric function values for half an angle based on the function values for the full angle. They are derived from the double-angle formulas, with common forms such as sin(?/2) = ±?[(1 - cos?)/2] and cos(?/2) = ±?[(1 + cos?)/2]. Determining the appropriate sign requires knowledge of the quadrant in which the half-angle lies.

Quadrant and Sign Analysis

Quadrant and sign analysis is crucial when dealing with trigonometric functions because the sign of sine, cosine, and tangent depends on the angle’s location in the coordinate plane. For instance, in the third quadrant both sine and cosine are negative, while for a half-angle residing in the second quadrant, sine is positive and cosine is negative. Correct sign assignment ensures accurate calculations when using identities.

Pythagorean Identity

The Pythagorean identity, stated as sin²? + cos²? = 1, is a fundamental relation in trigonometry that allows one to determine the sine or cosine of an angle when the other is known. It is especially useful when a value is provided (for example, cos?) and the corresponding sine value is needed, with the correct sign determined by the quadrant in which the angle lies.

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