Chapter Five: Analytic Trigonometry

5.1 Using Fundamental Identities

The Fundamental Trigonometric Identities include:

Reciprocal Identities

1

sincsc

uu

1

cossec

uu

1

tancot

uu

1

cscsin

uu

1

seccos

uu

1

cottan

uu

Quotient Identities

sin

tancos

uu

u

coscot

sin

uu

u

Pythagorean Identities

2 2sin cos 1u u 2 21 tan secu u 2 21 cot cscu u

Co-function Identities

sin cos2u u

cos sin

2u u

tan cot

2u u

csc sec2u u

sec csc

2u u

cot tan

2u u

Even/Odd Identities

sin sinu u cos cosu u tan tanu u

csc cscu u sec secu u cot cotu u

Using only these identities we can do a lot with trig functions.

Examples: Use the given values to evaluate (if possible) all six trig functions.

1. 3 3

tan , cos3 2

x x

2. 3 3 5

sec , csc2 5

Ex: Simplify the trigonometric expression.

1. tan cscx x

2. 21 cos cscx x

3. 4 4sec tanx x

4. sin cotx x

5. csc

sec

6. sin tan cos