Title: College Trigonometry 2 Credit hours through KCKCC or Donnelly
1Chapter 5Trigonometric Identities
Section 5.1 Fundamental Identities
Section 5.2 Verifying Identities
Section 5.3 Cos Sum and Difference
Section 5.4 Sin Tan Sum and Dif
Section 5.5 Double-Angle Identities
Section 5.6 Half-Angle Identities
2Section 5.1 Fundamental Identities
- Review of basic Identities
- Negative-Angle Identities
- Fundamental Identities
3sin ? cos ?
tan ?
Hypotenuse r
opposite side y
?
A
adjacent side x
4csc ? sec ?
cot ?
B
Hypotenuse r
opposite side y
?
A
C
adjacent side x
5The Reciprocal Identities
1 csc
1 sin
1 sec
1 cos
1 cot
1 tan
6The quotient Identities
cos sin
7The Negative-Angle Identities
- sin(-) - sin
- cos(-) cos
- tan(-) - tan
8This is our first Pythagorean identity
9Pythagorean identities
cos2? sin2? 1 or 1 tan2?
sec2? or tan2? 1 sec2?
cos2?
cos2?
cos2?
10Pythagorean identities
cos2? sin2? 1 or cot2? 1
csc2? or 1 cot2? csc2?
sin2?
sin2?
sin2?
11Section 5.2 Verifying Identities
- Verify Identities by Working with One Side
- Verify Identities by Working with Two Sides
12Hints for Verifying Identities
- Learn the fundamental identities and their
equivalent forms. - Simplify using sin and cos.
- Keep in mind the basic algebra applies to trig
functions. - You can always go down to x, y, and r
13Section 5.3 Cos Sum Difference
- Difference Identity for Cosine
- Sum Identity for Cosine
- Co-function Identities
- Applying the Sum and Difference Identities
14Cosine of the Sum or Difference
- cos(A B) cos A cos B sin A sin B
- cos(A - B) cos A cos B sin A sin B
15Co-function Identities
- sin (90à - à) cos à
- cos (90à - à) sin à
- tan (90à - à) cot à
- csc (90à - à) sec à
- sec (90à - à) csc à
- cot (90à - à) tan à
16Section 5.4 Sine and TangentSum and Difference
Identities
- Sum Identity for Sine
- Difference Identity for Sine
- Applying the Sum and Difference Identities for
Sine
17Sine of the Sum or Difference
- sin(A B) sin A cos B cos A sin B
- sin(A - B) sin A cos B - cos A sin B
18Tangent of the Sum or Difference
tan A tan B 1 tan A tan B
tan A - tan B 1 tan A tan B
19Section 5.5 Double-Angle Identities
- Double-Angle Identities
- Verifying Identities with Double Angels
- Applying Double-Angle Identities
20Double-Angle Identity Cosine
- cos(2A) cos(AA)
- cos A cos A sin A sin A
- cos2 A sin2 A
- or
- cos(2A) cos2 A sin2 A
- (1 - sin2 A) sin2 A
- 1 - 2sin2 A or 2cos2 A - 1
21Double-Angle Identity Sine
- sin(2A) sin(AA)
- sin A cos A cos A sin A
- 2sin A cos A
22Double-Angle Identity Tangent
tan A tan A 1 tan A tan A
2 tan A 1 tan2A
23Section 5.6 Half-Angle Identities
- Half-Angel Identities
- Using the Half-Angle Identities
24Half-Angle Identity Sine
- cos 2A 1 - 2sin2 A
- -cos 2A -cos 2A
- 0 1 - 2sin2 A cos 2A
- - 2sin2 A -2sin2 A
- -2sin2 A 1 cos 2A
- sin2 A (cos 2A 1)
- 2
25Half-Angle Identity Sine (cont.)
A 2
26Half-Angle Identity Cosine
- cos 2A 2cos2 A - 1
- 1 1
- cos 2A 1 2cos2 A
- 2cos2 A 1 cos 2A
- cos2 A (1 cos 2A)
- 2
27Half Angle Identity Cosine (cont.)
A 2
28Half-Angle Identity Tangent
A 2
sin
A 2
A 2
cos
ñ
A 2
1 cos A 1 cos A
29Half-Angle Identity Tangent (cont)
A 2
A 2
A 2
sin
2sin cos
A 2
A 2
cos
A 2
2cos2
( )
A 2
sin 2 sin A
A 2
( )
A 2
1 2cos 1 cos A
30Half-Angle Identity Tangent (cont)
- Using the other formula we get
- tan
-
A 2