College Trigonometry 2 Credit hours through KCKCC or Donnelly

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Chapter 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
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Section 5.1 Fundamental Identities

• Review of basic Identities
• Negative-Angle Identities
• Fundamental Identities

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sin ? cos ?
tan ?
Hypotenuse r
opposite side y
?
A
adjacent side x
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csc ? sec ?
cot ?
B
Hypotenuse r
opposite side y
?
A
C
adjacent side x
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The Reciprocal Identities

• sin csc
• cos sec
• tan cot

1 csc
1 sin
1 sec
1 cos
1 cot
1 tan
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The quotient Identities

• tan
• cot

cos sin
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The Negative-Angle Identities

• sin(-) - sin
• cos(-) cos
• tan(-) - tan

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This is our first Pythagorean identity
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Pythagorean identities
cos2? sin2? 1 or 1 tan2?
sec2? or tan2? 1 sec2?

cos2?
cos2?
cos2?
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Pythagorean identities
cos2? sin2? 1 or cot2? 1
csc2? or 1 cot2? csc2?

sin2?
sin2?
sin2?
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Section 5.2 Verifying Identities

• Verify Identities by Working with One Side
• Verify Identities by Working with Two Sides

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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

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Section 5.3 Cos Sum Difference

• Difference Identity for Cosine
• Sum Identity for Cosine
• Co-function Identities
• Applying the Sum and Difference Identities

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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

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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

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Section 5.4 Sine and TangentSum and Difference
Identities

• Sum Identity for Sine
• Difference Identity for Sine
• Applying the Sum and Difference Identities for
  Sine

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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

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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
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Section 5.5 Double-Angle Identities

• Double-Angle Identities
• Verifying Identities with Double Angels
• Applying Double-Angle Identities

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Double-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

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Double-Angle Identity Sine

• sin(2A) sin(AA)
• sin A cos A cos A sin A
• 2sin A cos A

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Double-Angle Identity Tangent

• tan 2A tan (A A)

tan A tan A 1 tan A tan A
2 tan A 1 tan2A
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Section 5.6 Half-Angle Identities

• Half-Angel Identities
• Using the Half-Angle Identities

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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

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Half-Angle Identity Sine (cont.)

• sin A
• sin

A 2
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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

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Half Angle Identity Cosine (cont.)

• cos A
• cos

A 2
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Half-Angle Identity Tangent

• tan
• tan

A 2
sin
A 2
A 2
cos
ñ
A 2
1 cos A 1 cos A
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Half-Angle Identity Tangent (cont)

• tan
• tan

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
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Half-Angle Identity Tangent (cont)

• Using the other formula we get
• tan

A 2