0. Functions

Using the Addition Formulas

Use the addition formulas to derive the identities in Exercises 31–36.

cos (x − π/2) = sin x

What happens if you take B = 2π in the addition formulas? Do the results agree with something you already know?

Apply the formula for cos (A − B) to the identity sin θ = cos (π/2 − θ) to obtain the addition formula for sin (A + B).

Prove the following identities.

$\(\frac{\sin\theta}{1+\cos\theta}\)=\(\frac{1-\cos\theta}{\sin\theta}\)$

Prove the following identities.

$\(\tan\)2\(\theta\)=\(\frac{2\tan\theta}{1-\tan^2\theta}\)$

Simplify the expression.

$\(\tan\)^2\(\theta\)-\(\sec\)^2\(\theta\)+1$

The law of sines The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, then

(sin A) / a = (sin B) / b = (sin C) / c

Use the accompanying figures and the identity sin (π − θ) = sin θ, if required, to derive the law.

Find all solutions to the equation where 0 ≤ $\(\theta\)$ ≤ $2\(\pi\)$.

$\(\sin\]\theta\[\cos\]\left\)(2\(\theta\[\right\))-\(\sin\]\left\)(2\(\theta\[\right\))\(\cos\]\theta\)=\(\frac{\sqrt2}{2}\)$

Simplify the expression.

$\(\left\)(\(\frac{\tan^2\theta}{\sin^2\theta}\)-1\(\right\))\(\csc\)^2\(\left\)(\(\theta\[\right\))\(\cos\)^2\(\left\)(-\(\theta\]\right\))$

For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.

cos 2θ + cos θ = 0

Evaluate sin (5π/12).

Using the Addition Formulas

Use the addition formulas to derive the identities in Exercises 31–36.

sin (x − π/2) = −cos x

Identify the most helpful first step in verifying the identity.

$\(\left\)(\(\frac{\tan^2\theta}{\sin^2\theta}\)-1\(\right\))=\(\sec\)^2\(\theta\[\sin\)^2\(\left\)(-\(\theta\]\right\))$