Download 5.5 Notes

5.5: Multiple Angle
December 1, 2016
Double Angle Formulas
Using double angle with x, y and r
Find the values of sin 2θ, cos 2θ, and tan 2θ for the
given value and interval.
1. sin θ =
12
,
13
2. tan θ =
1
2
,
(0°, 90°)
3𝜋
𝜋,
2
Try it Out
Find the values of sin 2θ, cos 2θ, and tan 2θ for the
given value and interval.
1. cos θ =
2
,
5
𝜋
− ,0
2
2. tan θ = – 3 ,
𝜋
,𝜋
2
Solving when x has coefficients
• There was a coefficient with the x – which
changes the number of answers we get
• With a coefficient of 1 – our answers come from
[0, 2𝜋]
• The larger the number, it expands our domain:
EX: 2x doubles our domain, we get 2 times the
answers
• The smaller the number, it shortens our domain:
1
EX: 𝑥 – our domain gets cut in half
2
Solving on the interval [0, 2𝜋]
1
1.) sin 2  
2
3
2.) cos 3 
2
1
3.) tan   1
2
Solving on the interval [0, 2𝜋]
1.) 4sin 2  2
3
2.)  3cos   3
2
3.) tan 2  1
Solving Equations with Multiple
Angles
Use an identity to solve each equation, correct to the
nearest degree, on the interval [0°, 360°].
1. cos 2x = sin x
2. sin 2x = sin x
3. cos 2x + cos x + 1 = 0
Solving Multiple Angles
Use an identity to solve each equation on the interval [0, 2π].
1. 2cos2 x – sin x – 1 = 0
2. cos2 x – 2sin x – 2 = 0
3. 4sin2 x = 5 – 4cos x
Half Angle Formulas
Practice
Find the exact value of cos 112.5
Find the exact value of sin 157.5
Practice with x, y, and r