5.1 Using the Fundamental

Identities

1) Evaluate Trig Functions

2) Simplify Trig Expressions

3) Develop Additional Trig Identities

4) Solve Trig Equations

4 Main Goals:

Reciprocal Identities

Fundamental Trig Identities (page 354)

Csc1 Sin

Sec1 Cos

Cot 1 Tan

Sin 1 Csc

Cos1 Sec

Tan 1 Cot

Quotient Identities

Fundamental Trig Identities (page 354)

Cos

Sin Tan

Sin Cos Cot

Pythagorean Identities

Fundamental Trig Identities (page 354)

1 CosSin 22 SecTan 1 22 CscCot 1 22

Cofunction Identities

Fundamental Trig Identities (page 354)

Cos )2

(Sin Sin )2

( Cos

Sec )2

( Csc Csc )2

( Sec

Cot )2

(Tan Tan )2

(Cot

Even/Odd Identities

Fundamental Trig Identities (page 354)

Sin - )(Sin Cos )( Cos Tan - )(Tan

Csc- )( Csc Sec )( Sec Cot - )(Cot

Use the values and Tan θ > 0 to find the values of all 6 functions.

Using the identities23- Sec

Sec1 Cos

32-

1 CosSin 22 1 )

32 (-Sin 22

95 Sin 2

35- Sin

Use the values and Tan θ > 0 to find the values of all 6 functions.

Using the identities23- Sec

32 - Cos

35- Sin

53 - Csc

32

35

25

Cos

Sin Tan

Csc x = 4; Cos x < 0

Using the identities

41 Sin 1 Cos Sin 22

1 Cos 41 2

2

1615 Cos2

415- Cos

Csc x = 4; Cos x < 0

Using the identities

41 Sin

Cos

Sin Tan

415- Cos

154- Sec 4

154

1 Tan

151-

Tan θ is undefined, Sin θ > 0

Tan θ is undefined → Cos θ = 0

Using the Identities

1 Cos Sin 22

1 0 Sin 22 1 Sin

1 Sin

Tan θ is undefined, Sin θ > 0

Sin θ = 1 Cos θ = 0 Tan θ is undef.

Using the Identities

Csc θ = 1 Sec θ = undef. Cot θ = 0

Using the identities

33 Tan

23 - Cos

21- Sin

2- Csc 3

32 - Sec

3 Cot

To simplify a trig expression means to reduce it to simplest term

This typically means reducing a larger expression to 1 trig function

Never want any fractions in our answer (reciprocal identities)

Simplifying Trig Expressions

Simplify the following expression:

(Cot x) )(Sin x

)Sin x

xCos( )(Sin x xCos

Simplify the following expression:

x)(Cos )Tan - (1 2x

x)(Cos )(Sec2x

xSec1 x)(Sec2 xSec

Simplify the following expression:

xCos

x)- 2

(Cos2

xCosSin 2x Sin x

xCosSin

x

Sin xTan x

xSin xCos

1 2 xSin x Sec 2

5.1 Using the Fundamental

Identities

Homework Even Answers

3 Cot x 3

32 - x Sec

2- x Csc 2

1- Sin x 2)

34 Cot x

45 x Sec

54 x Cos 5

3 Sin x 4)

310 - x Sec

10 x Csc 3

1- Tan x 10

103 - x Cos 6)

Homework Even Answers

34 Cot x

45 x Sec

35 x Csc

43 Tan x 5

3 Sin x 8)

1515 Cot x

154 x Csc

15 Tan x 4

1 x Cos 4

15 Sin x 10)

62 Cot x 12

65 - x Sec

126 Tan x

562 - x Cos

51 - Sin x 12)

Homework Even Answers

0 Cot x undef. x Sec 1 x Csc 0 x Cos 1 Sin x 14)

c 20)f 18)a 16)

Match the expressions to one of the following:

xSecSin x 1)

1) -x (Secx Cos 2) 22

xTan -x Sec 3) 44

xSecCot x 4)

xSin1 -x Sec 5) 2

2

xCsc a)Tan x b)

xSin c) 2

xSec d) 2

xTanx Sec e) 22

As we continue through the chapter, the problems with increase in difficulty

Always try to use the identities when possible

Last Resort is to convert all to sines and cosines◦ A common mistake is starting all problems by

converting all to sines and cosines. Do this last!

Keep in mind:

5.1 Using the Fundamental

Identities

So far, all the problems we have done have involved using the identities

Now, your first step should be to look to factor, then try to use the identities

What do you know how to factor?

Factoring

1) Factor out a termSin x Cos² - Sin xSin x (Cos²x – 1)

2) Factor a trinomialSin²x - 5Sin x + 6(Sin x – 2) (Sin x – 3)

3) Factor special polynomialsSin³x - Sin²x – Sin x + 1(Sin²x – 1) (Sin x - 1)

Factoring

Sin x Cos²x – Sin x

Can we factor?

Sin x (Cos²x – 1)Sin x (Sin²x)Sin³ x

Simplify the following

Simplify the following

2 - x Cos4 -x Cos2

2 - x Cos2) - x (Cos 2) x (Cos

2 x Cos

Simplify the following

3 -Tan x x Tan4 2

4x² + x - 3

If you get stuck, let x = Tan x

= (2x + 3) (x – 1)

1) -(2Tan x 3) Tan x (2

Simplify the following

xCos x Cos21 42 x)Cos - (1 x)Cos - (1 22

xSinx Sin 22

xSin 4

Simplify the following

1 x Sec -x Sec -x Sec 23

1) - x (Sec 1) -x (Sec 2

1) - x (Secx Tan 2