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
Chapter 5Trigonometric Identities
Section 5.6 Half-Angle Identities
Section 5.1 Fundamental Identities
• Review of basic Identities
• Negative-Angle Identities
• Fundamental Identities
sin θ =
cos θ =
tan θ =
yr
xr
yx
A
op
po
site side =
y
Hypot
enus
e =
r
adjacent side = x
θ
csc θ =
sec θ =
cot θ =
ry
rx
xy
A
opposite side = y
Hypot
enus
e =
r
adjacent side = x
B
Cθ
The Reciprocal Identities
sin £ = csc £ =
cos £ = sec £ =
tan £ = cot £ =
1csc £
1sec £
1cot £
1sin £
1cos £
1tan £
The quotient Identities
tan £ = =
cot £ = =
sin £cos £
cos £sin £
yx
xy
The Negative-Angle Identities
sin(-£) = - sin £ cos(-£) = cos £ tan(-£) = - tan £
x2 + y2 = r2
or
cos2θ + sin2θ = 1
r2 r2 r2
x
y
r
θ This is our first
Pythagorean identity
cos2θ + sin2θ 1
or
1 + tan2θ = sec2θor
tan2θ + 1 = sec2θ
x
y
r
θ
Pythagorean identities
cos2θ =cos2θ cos2θ
cos2θ + sin2θ 1
or
cot2θ + 1 = csc2θor
1 + cot2θ = csc2θ
x
y
r
θ
Pythagorean identities
sin2θ sin2θ sin2θ=
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 TangentSum 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 B1 – tan A tan B
tan A - tan B1 + 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 2A = tan (A + A) =
=
tan A + tan A1 – 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 =
sin =
‘ñ 1 – cos 2A 2
‘ñ 1 – cos A 2
A2
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 =
cos =
‘ñ 1 + cos 2A 2
‘ñ 1 + cos A 2
A2
Half-Angle Identity Tangent
tan = =
tan =
A2
sin
cos
A2A2
‘ñ 1 – cos A 2
ñ 1 + cos A 2
A2 ‘ñ1 – cos A
1 + cos A
Half-Angle Identity Tangent (cont)
tan = =
tan = =
A2
sin
cos
A2A2
A2
A2
A2
2sin cos
2cos2
A2
sin 2 sin A
1 + 2cos 1 + cos AA2
( )( )
A2
Half-Angle Identity Tangent (cont)
Using the other formula we get:
tan =
A2 sin A
1 - cos A
Top Related