Evaluate Expressions Involving Double Angles

If on the interval , find sin 2θ, cos

2θ, and tan 2θ.

Since on the interval , one point

on the terminal side of θ has y-coordinate 3 and a

distance of 4 units from the origin, as shown.

Evaluate Expressions Involving Double Angles

Use these values and the double-angle identities for sine and cosine to find sin 2θ and cos 2θ.

Then find tan 2θ using either the tangent double-angle identity or the definition of tangent.

sin 2θ = 2sin θ cos θ cos 2θ = 2cos2θ – 1

The x-coordinate of this point is therefore or .

Using this point, we find that cos θ = and

tan θ = .

Evaluate Expressions Involving Double Angles

Method 1

Evaluate Expressions Involving Double Angles

Method 2

Evaluate Expressions Involving Double Angles

Answer:

Solve an Equation Using a Double-Angle Identity

Solve cos 2θ – cos θ = 2 on the interval [0, 2π).

cos 2θ – cos θ

= 2

Original equation

2 cos2 θ – 1 – cos θ – 2

= 0

Cosine Double-Angle Identity

2 cos2 θ – cos θ – 3

= 0

Simplify.

(2 cos θ – 3)( cos θ + 1)

= 0

Factor.

2 cos θ – 3 = 0 or cos θ + 1

= 0

Zero Product Property

cos θ = or cos θ

= –1

Solve for cos θ.

θ

= π

Solve for

Solve an Equation Using a Double-Angle Identity

Answer: π

Since cos θ = has no solution, the solution on the

interval [0, 2π) is θ = π.

Use an Identity to Reduce a Power

Rewrite csc4 θ in terms of cosines of multiple angles with no power greater than 1.

csc4 θ = (csc2 θ)2 (csc2 θ)2 = csc4 θ

Reciprocal Identity

Pythagorean Identity

Cosine Power-Reducing Identity

Use an Identity to Reduce a Power

Common denominator

Simplify.

Cosine Power-Reducing Identity

Square the fraction.

Use an Identity to Reduce a Power

Simplify.

Common denominator

So, csc4 θ = .

Answer:

Solve an Equation Using a Power-Reducing Identity

Solve sin2 θ + cos 2θ – cos θ = 0.

Solve Algebraically

sin2 θ + cos 2θ – cos θ = 0 Original equation

Sine Power-Reducing Identity

Add like terms.

Double-Angle Identity

Simplify.

Multiply each side by 2.

Solve an Equation Using a Power-Reducing Identity

Factor.

2cos θ = 0 cos θ – 1= 0 Zero

Product Property

cos = 0 cos = 1 Solve for

cos .

= = 0 Solve for θ on [0, 2π).

The graph of y = sin2 θ + cos 2θ – cos θ has a period of 2, so the solutions are

Solve an Equation Using a Power-Reducing Identity

Support Graphically

The graph of y = sin2 θ + cos 2θ – cos θ has zeros at

on the interval [0, 2π).

Answer:

Evaluate an Expression Involving a Half Angle

Find the exact value of sin 22.5°.

Notice that 22.5° is half of 45°. Therefore, apply the half-angle identity for sine, noting that since 22.5° lies in Quadrant I, its sine is positive.

Sine Half-Angle Identity (Quadrant I angle)

Evaluate an Expression Involving a Half Angle

Quotient Property of Square Roots

Subtract and then divide.

Answer:

Evaluate an Expression Involving a Half Angle

CHECK Use a calculator to support your assertion

that sin 22.5° = . sin 22.5° = 0.3826834324

and = 0.3826834324

Solve an Equation Using a Half-Angle Identity

Solve on the interval [0, 2π).

Subtract 1 – cos x from each side.

Square each side.

Sine and Cosine Half-Angle Identities

Original equation

Multiply each side by 2.1 – cos x = 1 + cos x

Solve an Equation Using a Half-Angle Identity

Solve for cos x.

Answer:

Solve for x.

The solutions on the interval [0, 2π) are .