Download Ch 7 – Trigonometric Identities and Equations

7.1 – Basic
Trigonometric
Identities and
Equations
Trigonometric Identities
Quotient Identities
sin 
tan  
cos 
cos 
cot  
sin 
Reciprocal Identities
1
sin  
csc 
1
cos  
sec 
1
tan  
cot 
Pythagorean Identities
sin2 + cos2 = 1
tan2 + 1 = sec2
cot2 + 1 = csc2
sin2 = 1 - cos2
tan2 = sec2 - 1
cot2 = csc2 - 1
cos2 = 1 - sin2
5.4.3
Where did our pythagorean identities come from??
Do you remember the Unit Circle?
• What is the equation for the unit circle?
x2 + y2 = 1
• What does x = ? What does y = ?
(in terms of trig functions)
sin2θ + cos2θ = 1
Pythagorean
Identity!
Take the Pythagorean Identity and
discover a new one!
Hint: Try dividing everything by cos2θ
sin2θ + cos2θ = 1 .
cos2θ cos2θ cos2θ
tan2θ + 1 = sec2θ
Quotient
Identity
another
Pythagorean
Identity
Reciprocal
Identity
Take the Pythagorean Identity and
discover a new one!
Hint: Try dividing everything by sin2θ
sin2θ + cos2θ = 1 .
sin2θ sin2θ sin2θ
1 + cot2θ = csc2θ
Quotient
Identity
a third
Pythagorean
Identity
Reciprocal
Identity
Using the identities you now know,
find the trig value.
1.) If cosθ = 3/4, find secθ
2.) If cosθ = 3/5, find cscθ.
sin 2   cos 2   1
1
1
4
sec  


cos 3
3
4
2


3
sin 2      1
 5
25 9
sin 2  

25 25
16
2
sin  
25
4
sin   
5
csc  
1
1
5


sin   4
4
5
3.) sinθ = -1/3, find tanθ
1  cot 2   csc 2 
1  cot 2   (3) 2
cot 2   8
2
tanq = ±
4
cot 2   8
4.) secθ = -7/5, find sinθ
cos 2 q + sin 2 q = 1
( 7)
-5
25
49
2
+ sin 2 q = 1
+ sin 2 q = 1
24
sin q =
49
2
2 6
sin q = ±
7
REMEMBER….
TO NUMBER EACH STEP
WRITE CLEARLY
GO ALL THE WAY TO ONE TRIG VALUE
(DON’T LEAVE TAN2X, LEAVE TANX)
Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify.
a)
cos   sin  tan 
sin
 cos   sin 
cos 
sin2 
 cos  
cos 
cos   sin 

cos 
1

cos 
2
 sec 
2
cot 2 
1  sin2 
b)
cos 2 
2
sin


cos 2 
1
cos 2 
1


2
sin  cos2 

1
2
sin 
 csc 2 
5.4.5
Simplifing Trigonometric Expressions
c)
(1 + tan
x)2
- 2 sin x sec x
1
cos x
sin x
2
 1  2 tan x  tan x  2
cos x
 (1  tan x)  2 sin x
2
 1  tan2 x  2tanx  2 tanx
 sec2 x
d)
csc x
tan x  cot x
1

sin x
sin x cos x

cos x sin x
1

sin x
sin 2 x  cos 2 x
sin xcos x
1

sin x
1
sin x cos x
1
sin x cos x


sin x
1
 cos x
Simplify each expression.
1
 cos x 
cos x
  sin x
 sin x 
sin 
cos
sin 
 1  sin x 
 cos x


 sin x  cos x 
1
sin 

sin  cos 
1

1
 sec 
cos
cos 2 x sin 2 x

sin x
sin x
cos 2 x  sin 2 x
sin x
1
 csc x
sin x
Simplifying trig Identity
Example1: simplify
tanxcosx
sin
x
tanx cosx
cos x
tanxcosx = sin x
Simplifying trig Identity
Example2: simplify
sec x
csc x
1
cos
sec x
csc
1x
sin x
=
1
sinx
x
cos x
1
=
sin x
cos x
= tan x
Simplifying trig Identity
Example2: simplify
cos2x - sin2x
cos x
cos2x - sin
1 2x
cos x
= sec x
Example
Simplify:
= cot x (csc2 x - 1)
Factor out cot x
= cot x (cot2 x)
Use pythagorean identi
= cot3 x
Simplify
Example
Simplify:
= sin x (sin x) + cos x
cos x
cos
x
2
= sin x + (cos x)cos x
cos x
= sin2 x + cos2x
cos x
=
1
cos x
= sec x
Use quotient identity
Simplify fraction with
LCD
Simplify numerator
Use pythagorean iden
Use reciprocal identity
Your Turn!
Combine
fraction
Simplify the
numerator
Use
pythagorean
identity
Use Reciprocal
Identity
Practice
1
One way to use identities is to simplify expressions involving trigonometric
functions. Often a good strategy for doing this is to write all trig functions in
terms of sines and cosines and then simplify. Let’s see an example of this:
substitute using each
identity
sin x
tan x 
cos x
tan x csc x
Simplify:
sec x
simplify
sin x 1

 cos x sin x
1
cos x
1
 cos x
1
cos x
1
1
csc x 
sin x
1
sec x 
cos x
Another way to use identities is to write one function in terms of another
function. Let’s see an example of this:
Write the following expression
in terms of only one trig function:
cos x  sin x  1
2
= 1  sin 2 x  sin x  1
This expression involves both sine and
cosine. The Fundamental Identity makes a
connection between sine and cosine so we
can use that and solve for cosine squared
and substitute.
=  sin 2 x  sin x  2
sin 2 x  cos 2 x  1
cos 2 x  1  sin 2 x
(E) Examples
• Prove tan(x) cos(x) = sin(x)
LS  tan x cos x
sin x
LS 
cos x
cos x
LS  sin x
 LS  RS
21
(E) Examples
• Prove tan2(x) = sin2(x) cos-2(x)
RS  sin
2
x cos 2 x
1

2
RS  sin x  

2
cos
x


1
2
RS  sin x 
cos x 2
RS 
sin x 2
cos x 2
 sin x 
RS  

 cos x 
RS  tan 2 x
 RS  LS
2
22
(E) Examples
• Prove
tan x 
1
1

tan x sin x cos x
1
tan x
sin x
1

sin x
cos x
cos x
sin x
cos x

cos x
sin x
sin x sin x  cos x cos x
cos x sin x
2
sin x  cos2 x
cos x sin x
1
cos x sin x
 RS
LS  tan x 
LS 
LS 
LS 
LS 
LS 
 LS
23
(E) Examples
• Prove
LS
LS
LS
LS
sin 2 x
 1  cos x
1  cos x
sin 2 x

1  cos x
1  cos2 x

1  cos x
(1  cos x )(1  cos x )

(1  cos x )
 1  cos x
 LS  RS
24