Download 1 Simplifying Trig Expressions using Double and Half Angle Formulas

Simplifying Trig Expressions
using Double and Half Angle
Formulas
Lori Jordan
Kate Dirga
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Printed: November 16, 2013
AUTHORS
Lori Jordan
Kate Dirga
www.ck12.org
C ONCEPT
Concept 1. Simplifying Trig Expressions using Double and Half Angle Formulas
1
Simplifying Trig
Expressions using Double and Half
Angle Formulas
Here you’ll use the half and double angle formulas to simplify more complicated expressions.
As Agent Trigonometry, you are given the following cryptic clue. How could you simplify this clue?
tan 2x
tanx
1+tan x
Guidance
We can also use the double-angle and half-angle formulas to simplify trigonometric expressions.
Example A
Simplify
cos 2x
sin x cos x .
Solution: Use cos 2a = cos2 a − sin2 a and then factor.
cos 2x
cos2 x − sin2 x
=
sin x cos x
sin x + cos x
(
((
(sin
(cos
x+
x)
(cos x − sin x)(
((
=
(
(
sin(x(
+(
cos x
(
= cos x − sin x
Example B
Find the formula for sin 3x.
Solution: You will need to use the sum formula and the double-angle formula. sin 3x = sin(2x + x)
sin 3x = sin(2x + x)
= sin 2x cos x + cos 2x sin x
= 2 sin x cos x cos x + sin x(2 cos2 x − 1)
= 2 sin x cos2 x + 2 sin x cos2 x − sin x
= 4 sin x cos2 x − sin x
= sin x(4 cos2 x − 1)
We will explore other possibilities for the sin 3x because of the different formulas for cos 2a in the Problem Set.
Example C
Verify the identity cos x + 2 sin2 2x = 1.
1
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Solution: Simplify the left-hand side use the half-angle formula.
x
cos x + 2 sin2
2
!2
r
1 − cos x
cos x + 2
2
1 − cos x
2
cos x + 1 − cos x
cos x + 2 ·
1
Concept Problem Revisit
Use tan 2a =
2 tan a
1−tan2 a
and then factor.
2 tan x 1 + tan x
·
1 − tan2 x
tanx
2 tan x
1 + tan x
2
=
·
=
(1 + tan x)(1 − tan x)
tanx
1 − tan x
tan 2x
tanx
1+tan x
=
Guided Practice
1. Simplify
sin 2x
sin x .
2. Verify cos x + 2 cos2 2x = 1 + 2 cos x.
Answers
1.
sin 2x
sin x
=
2 sin x cos x
sin x
= 2 cos x
2.
x
cos x + 2 cos2 = 1 + 2 cos x
2
r
2
1 + cos x
cos x + 2
=
2
cos x + 1 + cos x =
1 + 2 cos x =
Practice
Simplify the following expressions.
√
1. 2 + 2 cos x cos 2x
2x
2. cos
cos2 x
3. tan 2x(1 + tan x)
4. cos 2x − 3 sin2 x
2x
5. 1+cos
cot x
6. (1 + cos x)2 tan 2x
Verify the following identities.
2
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7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Concept 1. Simplifying Trig Expressions using Double and Half Angle Formulas
sin x
cot 2x = 1−cos
x
1−cos x
sin x
=
1+cos x
sin x
sin 2x
=
tan
x
1+cos 2x
(sin x + cos x)2 = 1 + sin 2x
sin x tan 2x + 2 cos x = 2 cos2 2x
cot x + tan x = 2 csc 2x
cos 3x = 4 cos3 x − 3 cos x
cos 3x = cos3 x − 3 sin2 x cos x
sin 2x − tan x = tan x cos 2x
cos4 x − sin4 x = cos 2x
3