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Chapter 5
Trigonometric 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
Section 5.1 Fundamental Identities
• Review of basic Identities
• Negative-Angle Identities
• Fundamental Identities
sin θ =
y
r
x
r
tan θ =
y
x
A
θ
adjacent side = x
opposite side = y
cos θ =
csc θ =
r
y
r
x
x
y
A
θ
adjacent side = x
opposite side = y
sec θ =
cot θ =
B
C
The Reciprocal Identities
sin £ =
1
csc
£
=
csc £
1
sin £
cos £ =
1
sec
£=
sec £
1
cos £
tan £ =
1
cot
£ =£
cot
1
tan £
The quotient Identities
tan £ =
sin £
=
cos £
y
x
cot £ =
cos £
sin £
x
y
=
The Negative-Angle Identities
sin(-£) = - sin £
cos(-£) = cos £
tan(-£) = - tan £
x2 + y2 = r2
r 2 r2 r 2
or
r
y
θ
x
cos2θ + sin2θ = 1
This is our first
Pythagorean identity
Pythagorean identities
cos2θ + sin2θ
1
cos2θ cos2θ= cos2θ
or
1 + tan2θ = sec2θ
r
or
tan2θ
+1 =
y
sec2θ
θ
x
Pythagorean identities
cos2θ + sin2θ
1
= sin2θ
2
2
sin θ sin θ
or
cot2θ + 1 = csc2θ
r
or
1 +
cot2θ
=
y
csc2θ
θ
x
Section 5.2 Verifying Identities
• Verify Identities by Working with One Side
• Verify Identities by Working with Two
Sides
Hints 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
Section 5.3 Cos Sum & Difference
•
•
•
•
Difference Identity for Cosine
Sum Identity for Cosine
Co-function Identities
Applying the Sum and Difference Identities
Cosine 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
Co-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
Section 5.4 Sine and Tangent
Sum and Difference Identities
• Sum Identity for Sine
• Difference Identity for Sine
• Applying the Sum and Difference Identities
for Sine
Sine 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
Tangent of the Sum or Difference
tan (A + B) =
tan (A - B) =
tan A + tan B
1 – tan A tan B
tan A - tan B
1 + tan A tan B
Section 5.5 Double-Angle Identities
• Double-Angle Identities
• Verifying Identities with Double Angels
• Applying Double-Angle Identities
Double-Angle Identity Cosine
cos(2A) = cos(A+A)
= 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
Double-Angle Identity Sine
sin(2A) = sin(A+A)
= sin A cos A + cos A sin A
= 2sin A cos A
Double-Angle Identity Tangent
tan A + tan A
tan 2A = tan (A + A) =
1 – tan A tan A
2 tan A
= 1 – tan2A
Section 5.6 Half-Angle Identities
• Half-Angel Identities
• Using the Half-Angle Identities
Half-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
Half-Angle Identity Sine (cont.)
sin A =
‘ñ
1 – cos 2A
2
A
sin 2 =
‘ñ
1 – cos A
2
Half-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
Half –Angle Identity Cosine (cont.)
cos A =
‘ñ
1 + cos 2A
2
A
cos 2 =
‘ñ
1 + cos A
2
Half-Angle Identity Tangent
A
sin
2
A
tan 2 = cos A
2
A
tan 2 =
‘ñ
=
‘ñ
ñ
1 – cos A
1 + cos A
1 – cos A
2
1 + cos A
2
Half-Angle Identity Tangent (cont)
A
sin
2
A
tan 2 = cos A
2
A
tan 2 =
A
A
2sin 2 cos 2
=
()
()
A
sin 2
2
1 + 2cos A
2
2cos2 A
2
sin A
=
1 + cos A
Half-Angle Identity Tangent (cont)
Using the other formula we get:
1 - cos A
A
tan 2 =
sin A