Section 5.1 Fundamental 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

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