Download Lesson 12.2 Identities

Math-2 Honors
Lesson 12.2
Fundamental Identities
What you’ll learn about
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Identities
Basic Trigonometric Identities
Pythagorean Identities
Negative Angle Identities
Even/Odd Identities
Simplifying Trigonometric Expressions
Solving Trigonometric Equations
… and why
Identities are important when working with trigonometric
functions in calculus.
Trigonometric Functions
SOHCAHTOA
“Some old horse
caught another horse
taking oats away.”
opposite
sin A 
hypotenuse
o
sin A 
h
adjacent
cos A 
hypotenuse
a
cos A 
h
opposite
tan A 
adjacent
tan A 
o
a
SOH
CAH
TOA
Trigonometric Functions
Cosecant ratio
hypotenuse(length)
csc A 
opposite(length)
h
csc A 
o
o
sin A 
h
1
csc A 
sin A
Trigonometric Functions
Secant ratio
hypotenuse(length)
sec A 
adjacent (length)
h
sec A 
a
a
cos A 
h
sec A 
1
cos A
Trigonometric Functions
Cotangent ratio
adjacent (length)
cot A 
opposite (length)
a
cot A 
o
o
tan A 
a
Shot your cow:
1
cot A 
tan A
“Sha – Cho – Cao”
h
sec A 
a
h
csc A 
o
a
cot A 
o
Trigonometric Functions
Shot your cow:
“Sha – Cho – Cao”
h
sec A 
a
sin  csc
cos  sec
h
csc A 
o
a
cot A 
o
Unfortunately, ‘s’ doesn’t match
up with ‘s’ (or ‘c’ with ‘c’)
SHA-CHO-CAO
SOHCAHTOA
SOH
“shot your cow”
o
sin A 
h
CAH
a
cos A 
h
TOA
tan A 
1
csc A
1
csc A 
sin A
sin A 
h
sec A 
a
h
o
CHO
a
cot A 
o
CAO
csc A 
o
a
SHA
cos A 
1
sec A
1
tan A 
cot A
sec A 
1
cos A
1
cot A 
tan A
What is an Identity?
2(x – 3) = 2x – 6
x 1
 x 1
x 1
2
This is an equivalent expression.
It is true for all real numbers.
Like the equation above, it is a
true statement. BUT, it is only true
if x is in the domain of the expression
on the right AND left side.
When x = -1, it is not true.
Identity: an equation that is true for all values that are in
the domain of both sides of the equation.
Identity: an equation that is true for all values that are in
the domain of both sides of the equation.
Is it an Identity?
1.
3x  2  6x  4
3x  2  2(3x  2)
NOT an identity
2x  4x
2.
 2x
x2
2 x ( x  2)
 2x
x2
2
2x  2x
Domain:
left side: x ≠ 2
right side: all real #’s
NOT an identity
Basic Trigonometric Identities
Reciprocal Identities
1
sin  
csc
1
cos 
sec
1
tan  
cot
1
csc 
sin 
1
sec 
cos
1
cot 
tan 
Quotient Identities
sin 
tan  
cos
cos
cot 
sin 
Find the Product
sec Acos A
h
sec A 
a
a
cos A 
h
hyp adj
sec A cos A 
*
adj hyp
1
Find the Product/quotient
3. tan Acos A
4. cot Asin A
5.
tan A
sin A
6.
cot A
sec A *
* csc A
sin A
h
sec A 
a
h
csc A 
o
a
cot A 
o
a
cos A 
h
o
sin A 
h
o
tan A 
a
Using the UNIT CIRCLE
(x, y)
opp
y
sin  
 y
hyp
1
sin   y
r=1
y

x
Sine of the angle
is the vertical distance from
the x-axis to the point on the
circle.
Using the UNIT CIRCLE
(x, y)
r=1
y
adj  x  x
cos  
1
hyp
cos  x

x
Cosine of the angle
is the horizontal
distance from the y-axis
to the point on the circle.
Using the UNIT CIRCLE
The ordered pair (x, y) can
be re-written as:
(x, y) (cos  , sin  )
r=1

x
y
sin   y
cos  x
Using the UNIT CIRCLE
Using Pythagorean Theorem:
(cos  , sin  )
a2  b2  c2
x 2  y 2  12
(cos  ) 2  (sin  ) 2  1
r=1

x
y
We usually write (cos  ) 2
as: cos 2 
This give us our 1st
“Pythagorean” identity.
cos   sin   1
2
2
Pythagorean Identities
cos 2   sin 2   1
Divide both sides of the equation
by cos 2 
sin 2  cos 2 
1


2
2
cos  cos  cos 2 
This give us our 2nd
“Pythagorean” identity.
1  tan   sec 
2
2
Your turn:
3. Divide the first identity by sin 2  and simplify to
find the 3rd Pythagorean Identity.
1  cot   csc 
2
2
Solving equations
3x 1  5
To solve an equation we would use properties of equality
to “isolate the variable” on one side of the equal sign.
When using identities to solve equations we use
substitution.
Using Identities
3
Find sin if cos 
5
Which of these identities will help?
cos   sin   1
2
2
cot 2   1  csc 2 
1  tan   sec 
2
2
Substitution step.
2
3
2

sin
 1
 
5
9
 sin 2   1
25
9
sin   1 
25
2
25 9
sin  

25 25
2
25  9
sin  
25
2
16
sin  
25
4
sin  
5
2
Your Turn:
Given:
1
sin  
2
7.
cos
Find:
2
Given: cos  
2
8. Find: sin 
Using Identities
Find sin and cos  if tan  = 4
Which of these identities will help?
cos 2   sin 2   1
cot 2   1  csc 2 
1  tan   sec 
2
2
Substitution step.
1  4  sec 
2
2
1
sec  
cos 2 
2
1
17 
cos 2 
1
cos  
17
2
1
cos  
17
17
cos  
17
Using Identities
Find sin and cos  if tan  = 4
17
cos  
17
sin 
tan  
cos 
Which of these identities will help?
1
sec  
cos 
csc  
1
sin 
cot  
cos 
sin 
Substitution step.
sin 
4
17
17
17
4*
 sin 
17
4 17
sin  
17
Your Turn:
Given: sec(x) = 4,
9.
10.
Find:
tan(x)
Find: cot(x)
y
Angle A: sin A 
r
x
cos A 
r
x
Angle B: sin B 
r
y
cos B 
r
y
tan A 
x
x
cot A 
y
x
tan B 
y
y
cot B 
x
r
sec A 
x
r
csc A 
y
r
sec B 
y
r
csc B 
x
function of angle A = “cofunction” of angle B.
Negative Angle Identities
(“odd-even” identities) Sin θ = y coordinate of
the point on the circle.
sin(-θ) = -sin(θ)
(x, y)
“the y coord. of point
through which (-θ) passes
is the negative of the
y-coord of the point
through which (θ) passes.
θ
-θ
(x, -y)
Even-Odd Identities
cos θ = x coordinate of
the point on the circle.
(x, y)
Cos (-θ) = cos (θ)
“the x coord. of point
through which (-θ) passes
is the same as the
x-coord of the point
through which (θ) passes.
θ
-θ
(x, -y)
Even-Odd Identities
(x, y)
Sin (-θ) = -sin (θ)
1
csc  
sin 
csc (-θ) = - csc (θ)
θ
-θ
Cos (-θ) = cos (θ)
sec (-θ) = sec (θ)
(x, -y)
1
sec  
cos 
Even-Odd Identities
Sin (-θ) = -sin (θ)
sin 
tan  
cos 
Cos (-θ) = cos (θ)
(x, y)
tan(  ) 
sin(  )
cos(  )
 sin(  )
tan(  ) 
cos( )
θ
-θ
tan (-θ) = - tan (θ)
cot (-θ) = - cot (θ)
(x, -y)
Even-Odd Identities
sin(- x)  -sin x
cos(- x)  cos x
tan(- x)  - tan x
csc(- x)  - csc x
sec(- x)  sec x
cot(- x)  - cot x
The book uses ‘x’ instead of ‘θ’
for the angle variable.
Find: sin(  ) sec  
x
-x
1
  sin(  )
cos( )
  tan 
  sin(  ) sec 
 sin(  )

cos( )
Your Turn:
11.
sec( ) cos    ?
12.
tan(  ) cos    ?
Simplifying by Factoring and Using Identities
Simplify:
cot x sin x sec x  tan x cos x csc x
3
3
Try converting tan, cot, sec, csc into functions of sin and cos
cos
1
sin x
1
3
3
sin x

cos x
sin x
cos x cos x
sin x
cos x sin 2 x
From the properties
of exponents:
From the inverse
Property of multiplication:
From the Pythagorean
Property
1
1
 sin x cos 2 x
cos x
sin x
sin 2 x  cos 2 x
1
13. Simplify:
Your turn:
cot x tan x cos x cot x csc x  cot x tan x
cos x cos x
1
*
*
1
1
sin x sin x
2
cos x
1
2
sin x
cot 2  1
Pythagorean Identity
1  cot   csc 
2
 csc x
2
2
Simplifying expressions with Identities
sin 2 x, cos 2 x, tan 2 x, cot 2 x, sec 2 x, or csc 2 x
Anytime you see
refer to the Pythagorean Identities.
1  cos x
sin x
2
sin 2 x
sin x
cos   sin   1
2
2
1  tan   sec 
2
2
1  cot   csc 
2
2
sin x
sin   1  cos 
2
Use substitution
2
Your turn:
use identities to simplify
2
1
tan
x
13.
2
csc x
1  tan   sec 
2
sec 2 x
2
csc x
2
1 sin x
2
cos x 1
 tan 2 x
2
14.
sec x
 tan x
sin x
Convert into “sines” and “cosines”
1  sin x
 1
*


 cos x sin x  cos x
1


 cos x sin
 sin x sin x
*

x  cos x sin x
1  sin 2 x
cos x sin x
cos 2 x
cos x sin x
cos   sin   1
2
Need common denominator!
2
cos x
sin x
cot x
cos   1  sin 
2
2
15.
tan x tan x

2
2
csc x sec x
 sin x sin

*
 cos x 1
2
Convert into “sines” and “cosines”
  sin x cos x 
  

*
1 
  cos x
2
sin x(sin  cos x)
cos x
2
sin x * (1)
cos x
2
sin x
cos x
cos   sin   1
2
tan x
2