Download sum, difference and cofuntion identities

What you’ll learn about
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Identities
Basic Trigonometric Identities
Pythagorean Identities
Cofunction Identities
Odd-Even Identities
Simplifying Trigonometric Expressions
Solving Trigonometric Equations
… and why
Identities are important when working with
trigonometric
functions in calculus.
Basic Trigonometric Identities
Reciprocal Identites
1
1
csc 
sec 
sin 
cos
1
cot  
tan 
1
sin  
csc
1
tan  
cot 
1
cos 
sec
Quotient Identites
sin 
cos
tan  
cot 
cos
tan 
Pythagorean Identities
cos   sin   1
1  tan   sec 
cot   1  csc 
2
2
2
2
2
2
Example Using Identities
Find sin and cos if tan   3 and cos  0.
1  tan 2   sec 2 
To find sin , use tan   3
1  9  sec 2 
and cos  1 / 10.
sin 
tan  
cos
sin   cos tan 
sec   10
cos  1 / 10


sin   1 / 10 3
sin   3 / 10
Therefore, cos  1 / 10 and sin  3 / 10
Cofunction Identities
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
Cofunction Identities


sin      cos 
2



tan      cot 
2



sec      csc 
2



cos      sin 
2



cot      tan 
2



csc      sec 
2

Even-Odd Identities
sin(x)   sin x
csc(x)   csc x
cos(x)  cos x
sec(x)  sec x
tan(x)   tan x
cot(x)   cot x
Example Simplifying by
Factoring and Using Identities
Simplify the expression cos 3 x  cos xsin2 x.
Example Simplifying by
Factoring and Using Identities
Simplify the expression cos 3 x  cos xsin2 x.
cos x  cos x sin x  cos x(cos x  sin x)
 cos x(1)
Pythagorean Identity
 cos x
3
2
2
2
Example Simplifying by
Expanding and Using Identities
Simplify the expression:
csc x -1csc x  1
cos2 x
Example Simplifying by
Expanding and Using Identities
csc x  1csc x  1  csc 2 x  1
cos 2 x
(a  b)(a  b)  a 2  b 2
cos 2 x
cot 2 x

Pythagorean Identity
2
cos x
cos 2 x
1
cos


cot  
2
2
sin x cos x
sin 
1

sin 2 x
 csc 2 x
Example Solving a
Trigonometric Equation
Find all values of x in the interval 0,2 
sin 3 x
that solve
 tan x.
cos x
Example Solving a
Trigonometric Equation
3
sin x
 tan x
cos x
sin 3 x sin x

cos x cos x
3
sin x  sin x
Reject the posibility that cos 2 x  0
because it would make both
sides of the original equation
undefined. sin x  0 in the interval
0  x  2 when x  0 and x   .
sin 3 x  sin x  0
sin x(sin x  1)  0
2


sin x  0
or
2
sin
x
cos
x 0
 
cos 2 x  0
Quick Review
Evaluate the expression.
 4
1. sin  
 5
1
 12 
2. cos   
 13 
1
Factor the expression into a product of linear factors.
3. 2a 2  3ab  2b 2
4. 9u 2  6u  1
Simplify the expression.
2 3
5. 
y x
Quick Review Solutions
Evaluate the expression.
 4
1. sin  
 5
1
53.13o  0.927 rad
 12 
2. cos   
 13 
1
157.38o  2.747 rad
Factor the expression into a product of linear factors.
3. 2a 2  3ab  2b 2
4. 9u 2  6u  1
2a  b a  2b 
2
3u  1
Simplify the expression.
2 3
2x  3y
5. 
y x
xy
What you’ll learn about
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Cosine of a Difference
Cosine of a Sum
Sine of a Difference or Sum
Tangent of a Difference or Sum
Verifying a Sinusoid Algebraically
… and why
These identities provide clear
examples of how different
the algebra of functions can be from
the algebra of real
numbers.
Cosine of a Sum or
Difference
cos(u  v)  cos u cos v  sin u sin v
(Note that the sign does not switch in either case)
Example Using the Cosine-ofa-Difference Identity
Find the exact value of cos 75º without using a calculator.
cos 75º  cos 45º 30º 
 cos 45º cos 30º  sin 45º sin 30º
 2   3  2   1


 




2
2
2


 
 2

6 2
4
Sine of a Sum or
Difference
sin u  v   sinu cos v  sin v cosu
(Note that the sign does not switch in either case.)
Example Using the Sum
and Difference Formulas
Write the following expression as the sine or




cosine of an angle: sin cos  sin cos
3
4
4
3
Example Using the Sum
and Difference Formulas
Write the following expression as the sine or




cosine of an angle: sin cos  sin cos
3
4
4
3








Recognize sin cos  sin cos
as the sin(u  v).
3
4
4
3
 
sin cos  sin cos  sin   
 3 4
3
4
4
3
7
 sin
12
Tangent of a Difference of
Sum
sin(u  v) sin u cos v  sin v cos u
tan(u  v) 

cos(u  v) cos u sin v  sin u sin v
or
tan u  tan v
tan(u ) 
1  tan u tan v
Example Expressing a Sum
of Sinusoids as a Sinusoid
Find values for a, b, and h so that for all x,
4 cos 2x  7sin 2x  asin b x  h  .
Example Expressing a Sum
of Sinusoids as a Sinusoid
4 cos 2x  7sin 2x  asin b x  h 
2
is a sinusoid with period
  , so b  2
2
4 cos 2x  7sin 2x  asin  2 x  h 
 asin 2x  2h 
 asin 2x cos 2h  a cos 2x sin 2h
4 cos 2x  7sin 2x  asin 2h cos 2x  a cos 2h sin 2x,
so
4  asin 2h
and
 7  a cos2h
Example Expressing a Sum
of Sinusoids as a Sinusoid
4  asin 2h and
4
sin 2h  
and
a
4 a 4
so tan 2h 

7 a 7
so
 7  a cos 2h
7
cos 2h  
a
 4
so possible values of 2h are tan    n
 7
1
Example Expressing a Sum
of Sinusoids as a Sinusoid
4
To find a, we use sin 2h  
a
7
and cos 2h  
a
Because a is the hypotenuse,
a
7   4 
2
2
 65
Thus,
 
1 1  4    
4 cos 2x  7sin 2x  65 sin 2  x  tan     
 7  2 
2
 
Example Expressing a Sum
of Sinusoids as a Sinusoid
Support Graphically
The graphs of
y1  4 cos 2x  7sin 2x
and
 
1 1  4    
y2  65 sin 2  x  tan     
 7  2 
2
 
are identical.
Example Expressing a Sum
of Sinusoids as a Sinusoid
Interpret
1 1  4  
The values are a  65, b  2, and h  tan   
 7 2
2
(other values are possible).
The amplitude is 65, the peridod is  , and
1 1  4  
the phase shift is tan    radians.
 7 2
2
Quick Review
Express the angle as a sum or difference of special angles
(multiples of 30o, 45o,  /6, or  /4). Answers are not unique.
1. 15o
2.  /12
3. 75o
Tell whether or not the identity f (x  y)  f (x)  f (y)
holds for the function f .
4. f (x)  16x
5. f (x)  2e x
Quick Review Solutions
Express the angle as a sum or difference of special angles
(multiples of 30o, 45o,  /6, or  /4). Answers are not unique.
1. 15o
45o  30o
2.  /12



3 4
3. 75o
30o  45o
Tell whether or not the identity f (x  y)  f (x)  f (y)
holds for the function f .
4. f (x)  16x yes
5. f (x)  2e x
no
What you’ll learn about
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Double-Angle Identities
Power-Reducing Identities
Half-Angle Identities
Solving Trigonometric Equations
… and why
These identities are useful in calculus
courses.
Double Angle Identities
sin 2u  2sinu cosu
cos 2 u  sin 2 u

2
cos 2u  2 cos u  1
1  2sin 2 u

2 tanu
tan 2u 
2
1  tan u
Proving a Double-Angle
Identity
cos 2x  cos(x  x)
 cos x cos x - sin x sin x
 cos 2 x  sin 2 x
Power-Reducing Identities
1  cos 2u
sin u 
2
1  cos 2u
2
cos u 
2
1  cos 2u
2
tan u 
1  cos 2u
2
Example Reducing a
Power of 4
Rewrite sin 4 x in terms of trigonometric functions with
no power greater than 1.
Example Reducing a
Power of 4
sin x  sin x 
4
2
2
 1  cos 2x 



2
2
1  2 cos 2x  cos 2 2x

4
1 cos 2x 1  1  cos 4x 
 
 


4
2
4
2
1 cos 2x 1  cos 4x
 

4
2
8
Half-Angle Identities
u
1  cosu
sin  
2
2
u
1  cosu
cos  
2
2
 1  cosu

1

cosu

u 1  cosu
tan  
2  sinu
 sinu
1  cosu

Example Using a Double
Angle Identity
Solve cos x  cos 3x  0 in the interval [0,2 ).
Example Using a Double
Angle Identity
Solve cos x  cos 3x  0 in the interval [0,2 ).
Solve Graphically
The graph suggest that
there are six solutions:
0.79, 1.57, 2.36,
3.93, 4.71, 5.50.
Example Using a Double
Angle Identity
Solve cos x  cos 3x  0 in the interval [0,2 ).
Confirm Algebraically
cos x  cos 3x  0
cos x  cos x cos 2x  sin x sin 2x  0


cos x  cos x 1  2sin 2 x  sin x 2sin x cos x   0
cos x  cos x  2 cos x sin 2 x  2 cos x sin 2 x  0
2 cos x  4 cos x sin 2 x  0


2 cos x 1  2sin 2 x  0
Example Using a Double
Angle Identity
2 cos x 1  2sin x  0
2
cos x  0

3
x  or
2
2

3
x  or
2
2
or
or
or
1  2sin x  0
2
2
sin x  
2
 3 5
7
x  , , , or
4 4 4
4
The six exact solutions in the given interval are
 
3 5 3
7
, ,
,
,
, and
.
4 2 4 4 2
4
Quick Review
Find the general solution of the equation.
1. cot x  1  0
2. (sin x)(1  cos x)  0
3. cos x  sin x  0


4. 2sin x  2 2sin x  1  0
5. Find the height of the isosceles triangle with
base length 6 and leg length 4.
Quick Review Solutions
Find the general solution of the equation.
3
1. cot x  1  0
x
 n
4
2. (sin x)(1  cos x)  0
3. cos x  sin x  0
x  n
x

4
 n
Quick Review Solutions
Find the general solution of the equation.


4. 2sin x  2 2sin x  1  0
5
7
x
 2 n, x 
 2 n,
4
4

5
x   2 n, x 
 2 n
6
6
5. Find the height of the isosceles triangle with
base length 6 and leg length 4.
7
END OF THE LESSON 