Use the information given about the angle θ, 0 ≤θ<2 π, to find the exact value of: (a) sin(2 θ)

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Use the information given about the angle θ, 0 ≤θ<2 π, to...

Use the information given about the angle $\theta, 0 \leq \theta<2 \pi,$ to find the exact value of: (a) $\sin (2 \theta)$ (b) $\cos (2 \theta)$ (c) $\sin \frac{\theta}{2}$ (d) $\cos \frac{\theta}{2}$ (e) $\tan (2 \theta)$ (f) $\tan \frac{\theta}{2}$ $\csc \theta=-\sqrt{5}, \quad \cos \theta<0$

Use the information given about the angle $\theta, 0 \leq \theta<2 \pi,$ to find the exact value of:
(a) $\sin (2 \theta)$
(b) $\cos (2 \theta)$
(c) $\sin \frac{\theta}{2}$
(d) $\cos \frac{\theta}{2}$
(e) $\tan (2 \theta)$
(f) $\tan \frac{\theta}{2}$
$\csc \theta=-\sqrt{5}, \quad \cos \theta<0$

Key Concepts

Half-Angle Formulas

Half-angle formulas allow one to express the sine, cosine, and tangent of half an angle in terms of the cosine (or sine) of the original angle. They are derived from the double-angle formulas and take the form of square roots, as in sin(?/2) = ±?((1 ? cos?)/2) and cos(?/2) = ±?((1 + cos?)/2). The appropriate sign for these expressions is determined by the angle’s quadrant.

Double-Angle Formulas

Double-angle formulas express the trigonometric functions of twice an angle in terms of the functions of the original angle. For example, the formula for sine is sin(2?) = 2 sin? cos?, and for cosine, there are multiple forms such as cos(2?) = cos²? ? sin²? or cos(2?) = 2 cos²? ? 1. These formulas are widely used in problems that involve expressions with multiples of an angle.

Quadrant Analysis and Sign Determination

Trigonometric functions can be positive or negative depending on the quadrant in which the angle lies. Determining the correct sign for an expression—particularly when applying half-angle formulas—requires knowledge of the angle's quadrant. This concept is crucial when solving for trigonometric values since it ensures the resulting expressions have the correct sign according to standard conventions.

Trigonometric Identities

Trigonometric identities provide relationships between the trigonometric functions that hold true for any angle. These include fundamental identities like the Pythagorean identity, which relates sine and cosine, as well as formulas that allow the expression of multiple-angle functions such as sine, cosine, and tangent in terms of single-angle functions. Such identities are essential for simplifying expressions and solving equations involving trigonometric functions.

Pythagorean Identity

The Pythagorean identity, sin²? + cos²? = 1, is a key relation that allows one to compute one trigonometric function from another when one of them is known. This identity is instrumental when determining values of sine or cosine from given conditions, and it underpins many other trigonometric formulas and identities.

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