0. Functions

Solving Trigonometric Equations

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

sin² θ = cos² θ

Find all solutions to the equation.

$\(\left\)(\(\cos\[\theta\)+\(\sin\]\theta\[\right\))\(\left\)(\(\cos\]\theta\)-\(\sin\[\theta\]\right\))=-\(\frac\)12$

Identify the most helpful first step in verifying the identity.

$\(\sec\)^3\(\theta\)=\(\sec\]\theta\)+\(\frac{\tan^2\theta}{\cos\theta}\)$

Derive a formula for tan (A − B).

Using the Addition Formulas

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

sin (A − B) = sin A cos B − cos A sin B

Sag angle Imagine a climber clipping onto the rope described in Example 7 and pulling himself to the rope’s midpoint. Because the rope is supporting the weight of the climber, it no longer takes the shape of the catenary y = 200 cosh x/200. Instead, the rope (nearly) forms two sides of an isosceles triangle. Compute the sag angle θ illustrated in the figure, assuming the rope does not stretch when weighted. Recall from Example 7 that the length of the rope is 101 ft.

What are the three Pythagorean identities for the trigonometric functions?

Using the Half-Angle Formulas

Find the function values in Exercises 47–50.

sin² 3π/8

A triangle has side c = 2 and angles A = π/4 and B = π/3. Find the length a of the side opposite A.

Simplify the expression.

$\(\frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)}\)$

In Exercises 39–42, express the given quantity in terms of sin x and cos x.

cos (3π/2 + x)

Prove the following identities.

$\(\sec\]\theta\)=\(\frac{1}{\cos\theta}\)$

Evaluate cos (11π/12) as cos (π/4 + 2π/3).

Prove the following identities.

$\(\tan\]\theta\)=\(\frac{\sin\theta}{\cos\theta}\)$

Using the Half-Angle Formulas

Find the function values in Exercises 47–50.

cos² 5π/12