Pre-Calculus Honors
Book Reference 5.4
Unit 7 Lesson 11: Double Angle & Half Angle Formulas
Objective: _______________________________________________________
Do Now
One of your classmates is observing the triangle below. The classmate notices that sin  A 
3
.
5
The classmate then makes the prediction that sin 2  A 
6
. It this classmates prediction correct? If the
5
prediction is incorrect, explain why using the unit circle. Provide examples in your explanation.
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Communication
We will answer the question today of how to calculate double angles and half angles, given a right triangle.
1. Directions: Use the appropriate sum or difference identity to prove the angle identity sin 2  ?
2. Directions: Use the appropriate sum or difference identity to prove the angle identity cos 2  ?
3. Directions Mark up the following proof, so
you understand the concept of each algebraic
step.


To find cos(B/2), start with the cosine double angle
formula.
cos(2A) = 2cos ²A – 1
Use substitution to substitute <A= B/2:
cos (2 x B/2) = 2 cos²(B/2) – 1
cos (B) = 2 cos²(B/2) – 1

Rearrange and solve algebraically for cos(B/2):
cos( B)  1  2 cos 2 ( B / 2)
cos( B)  1
 cos 2 ( B / 2)
2
cos( B)  1
 cos( B / 2)
2
4. Directions Use the logic on the left to prove
the half angle formula for sin. Start with the
formula cos 2  1  2 sin 2  .
Double Angle and Half Angle Formula
Sine Half Angle Formula

1  cos 
sin 
2
2
Sine Double Angle Formula
sin 2  2 sin  cos
Cosine Double Angle Formulas
cos 2  cos 2   sin 2 
Cosine Half Angle Formulas

1  cos 
cos 
2
2
 2 cos   1
2
 1  2 sin 2 
Tangent Double Angle Formula
2 tan 
tan 2 
1  tan 2 
Tangent Half Angle Formula
q 1- cosq
sinq
tan =
=
2
sinq
1+ cosq
Find the exact value of the following using the diagram to the right. Do work on a separate sheet of paper if
necessary.
1.) sin 
2.) cos
3.) sin 2
4.) cos 2
5.) tan 2
6.) csc 2
7.) sin
9.) tan

2

2
8.) cos

2
10.) sec

2
Group Challenge Directions: Prove the following statements within your groups. If
all four members of your group get stuck you may ask for clues. If you ask for a
clue, you must record the clue the teacher gave you in the clue box below.
Clues:





#1 Prove the identity
Algebraic Proof
2 sin  cos3   2 sin 3  cos  sin 2
#2 Prove the identity
sin 3x  (sin x)(3  4 sin 2 x)
Verbal Steps
Unit 7 Lesson 11 Homework
Use the figure below to find the exact value of each trig function.
1.) sinq
2.) cosq
3.) cos2q
4.) tan2q
5.) sin

6.) cos
2
7.) tan

8.) csc
2
9.) 2 sin

2
cos

2

2

2
Use the appropriate sum or difference identity to prove the angle identity
2
10.) cos 2u  2 cos u  1
11.) cos 3x  cos x(4 cos 2 x  3)