November 5, 2012Using Fundamental Identities

Warm-up:Find the trig value for:1. sec(11π/6) 2. cot(2π/3) 3. csc(2π)

Find the angle θ for:4. tanθ = -√3 5.

6. cotθ = -1

csc 2 3

3

CW/HW 5.1: Pg. 379 #15-43, 45-53 Odds only

Derive the three Pythagorean Identities

1. sin2θ + cos2θ = 1

2. 1 + tan2θ = sec2θ

3. 1 + cot2θ = csc2θ

Lesson 5.1Using Fundamental Trig Identities

csc

1sin

sec

1cos

sin

1csc

cos

1sec

cot

1tan

tan

1cot

Reciprocal Identities

Quotient Identities

cos

sintan

cot cossin

Pythagorean Identities

sin2θ + cos2θ = 1 1+ tan2θ = sec2θ

1+ cot2θ = csc2θ

Cofunction Identities

sin2

cos

cos2

sin

tan2

cot

cot2

tan

sec2

csc

csc2

sec

Verify one of the cofunction identitiesShow that

Take a look at a 30-60-90 triangle 30°

60°

2

1

√3

cos60 sin(90 60 )

1

2sin(30 )

1

21

2✓

sin2

cos

Negative Angle Identities

)tan()tan(

)cos()cos(

)sin()sin(

Show that these identities are true. Use any angle for θ.

Simplifying an expressions to get a single value.

The goal is to use the identities to substitute and simplify. You want to try to get a single term.

Example 1: Transform the left side of the equation into the right side (0 < θ < π/2)

a) tanθ cotθ = 1 b) cotθ sinθ = cosθ

More simplifyingTry to rewrite as a single term

Example 2:a) b)

c)

sin csccot

1

tan2 1

cot2

x

cos x

Start practicing on HWPg. 379 #15-43

Use factoring to simplifyExample 3: common factor

difference of squaresa) sin2x csc2x – sin2x b) sec4x –

tan4x

Use these strategies for HW #45-53 odd

All the Trig Identities:

xx

xx

xx

22

22

22

csc1cot

sec1tan

1sincos

xx cos2

sin

xx sin2

cos

xx cot2

tan

xx

xx

xx

tan)tan(

cos)cos(

sin)sin(

x

xx

sin

coscot

x

xxcos

sintan

xxcos

1sec

xxsin

1csc