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5.1 Using the Fundamental Identities. 4 Main Goals:. Evaluate Trig Functions Simplify Trig Expressions Develop Additional Trig Identities Solve Trig Equations. Fundamental Trig Identities (page 354). Reciprocal Identities. Fundamental Trig Identities (page 354). Quotient Identities. - PowerPoint PPT Presentation
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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
5.1 Using the Fundamental
Identities
Use the substitution x = 2 Tan θ to express the following expression as a
trig function of θ
Trig Substitution
2 x 4
2 x 4 Tan 2 x
1. Substitute 2 Tan θ for x 2)Tan (2 4
2. Apply the rules for exponents Tan 4 4 2
3. Factor ) Tan (1 4 2
4. Simplify Sec 4 2 Sec 2
Simplify the following:a) x = 3 Sin θ in the expression
b) x = 2 Tan θ in the expression
c) x = 2 Cos θ in the expression
2 x- 9
25x2
216x - 64