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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 = .
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.
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