0. Functions

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

Using the Addition Formulas

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

cos (x − π/2) = sin x

Using the Addition Formulas

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

sin (x − π/2) = −cos x

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

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

cos (3π/2 + x)

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

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

Evaluate sin (5π/12).

Using the Half-Angle Formulas

Find the function values in Exercises 47–50.

cos² 5π/12

Using the Half-Angle Formulas

Find the function values in Exercises 47–50.

sin² 3π/8

Solving Trigonometric Equations

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

sin² θ = cos² θ

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.

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.