Math 4 Honors
Lesson 4-5: Double Angle Formulas
Name _______________________________
Date ____________________________
Learning Goal:

I can, without a calculator, use trigonometric identities such as angle addition/subtraction and
double angle formulas, to express values of trigonometric functions in terms of rational numbers and
radicals.
The Grand Finale . . .
The sum & difference formulas that we used in the previous investigations can be used to derive double
angle formulas. In this investigation we will derive a total of six double angle formulas for sine, cosine &
tangent.
A. There is one formula for sin (2x).
To derive a formula for sin (2x), think of sin (2x) as sin (x + x). Then use the sin (α + β) formula.
sin (2x) = sin (x + x) =
B. There are 3 formulas for cos (2x).
To derive a formula for cos (2x), use the same process you did in part A.
cos (2x) = cos (x + x) =
Now use your result from above to write 2 more formulas for cos (2x) – one in terms of just sine &
one in terms of just cosine.
cos (2x) =
cos (2x) =
C. There are 2 formulas for tan (2x).
To get the first one, “stack up” sin (2x) & cos (2x).
tan (2x) =
To derive the second one for tan (2x), use the same process you did in part A.
tan (2x) = tan (x + x) =
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Page 2
Transfer all six formulas you derived on the front of this page, below.
Call the Heinl over to your group for verification of your formulas.
cos 2 x 
sin 2 x 
cos 2 x 
tan 2 x 
cos 2 x 
tan 2 x 
Once you have Heinl approval, try examples 1 - 4. No calculator!
1. If
4
cos x  , find cos 2 x.
5
2. If sin x 
12

and  x   , find sin 2 x, cos 2 x and tan 2 x.
13
2
In what quadrant does 2x lie? How do you know?
3. Solve for primary values: 2cos x  sin 2x  0
4
4
4. Verify: cos x  sin x  cos 2 x
Page 3
Class examples: Solve for primary values.
5. 2sin(2x) + 1 = 0
U4 L1 I5 Homework:
5

1. Given : sin x  , 0  x 
13
2
2. Given :
cos x 
6. 2sin 2 (3x)  5sin(3x)  3
Find: sin 2x, cos 2x, tan 2x. In what quadrant does 2x lie?
5 3
,
 x  2 Find: sin 2x, cos 2x, tan 2x. In what quadrant does 2x lie?
3
2
3. Solve for primary values: sin 2x  cos x  0
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Page 4
4. Solve for primary values: 1  3cos x  cos 2x  0
5. Solve for primary values: sin 2 x  
3
2
6. Solve for primary values:
7. Verify:
1  cos 2 x
 cot x
sin 2 x
8. Verify:
9. Verify:
Hints: Work only on the left side &
.