CHAPTER 5
LESSON 5
Date_______________
Sum and Difference Identities
AW 5.5
MP 5.5
Objective:
• To use sum and difference identities for sine and cosine to verify and simplify
trigonometric expressions and to prove other identities.
Investigate
1. Consider the equation cos( A − B) = cos A − cos B
a) Verify the equation cos( A − B) = cos A − cos B numerically, using A =
B=
π
π
2
and
.
3
b) Is this an identity?
Left Side:
π π
cos( - ) =
2 3
π
cos
6
3
=
2
Right Side:
π
π
cos - cos =
2
3
1
02
1
=2
Left Side ≠ Right Side so the conjecture that cos(A - B) = cosA - cosB is not true.
2. Consider the equations
i. cos( A − B ) = cos A cos B − sin A sin B and
ii. cos( A − B) = cos A cos B + sin A sin B
a) Verify the equations numerically, using A =
π
2
and B =
π
3
.
b) Which equation appears to be true?
c) Guess and check an identity for cos( A + B )
π
π
The second conjecture works when A =
and B =
3
2
π π
π
3
cos( - ) = cos( ) =
2 3
6
2
cos
π
π
π
π
1
3
3
cos + sin sin = (0)( )+ (1)(
)=
2
3
2
3
2
2
2
Based on the same pattern as cos(A - B) = cosAcosB + sinAsinB then,
cos(A + B) = cosAcosB - sinAsinB can be verified by substitution; for example, if
π
A= π and B = ,
2
π
3π
=0
Left Side= cos(π + ) = cos
2
2
π
π
Right Side = cos π cos - sin π sin = (-1)(0) - (0)(1) = 0
2
2
3. a) Verify sin( x + y ) = sin x cos y + cos x sin y graphically by assigning y =
π
3
and
then graphing each side of the equation.
b) Guess and check a similar identity for sin( x − y )
π
.
3
Then the right side of the equation , sin x cos y + cos x sin y
π
π
= sin x cos + cos x sin
3
3
1
3
= sin x +
cos x
2
2
Let y =
π
1
3
) and y2 = sin x +
cos x appear to be identical so
3
2
2
sin(x + y) = sin x cos y + cos x sin y may be an identity.
The graphs of y1 = sin(x +
Based on the pattern above sin(x - y) = sin x cos y - cos x sin y . This conjecture tests
well via substitution or graphing.
The above explorations lead to the following sum and difference identities.
Sum and Difference Identities
sin( A + B) = sin A cos B + cos A sin B
sin( A − B) = sin A cos B − cos A sin B
cos( A + B ) = cos A cos B − sin A sin B
cos( A − B ) = cos A cos B + sin A sin B
tan( A + B) =
tan A + tan B
1 − tan A tan B
tan( A − B) =
tan A − tan B
1 + tan A tan B
Example 1:
Express the following as a trigonometric function of a single angle:
sin π cos
π
5
− cos π sin
π
5
.
This expression has the same pattern as sin(A - B) = sin A cos B - cos A sin B , so
π
π
sin π cos - cos π sin
5
5
π
= sin(π - )
5
4π
= sin
5
Example 2:
π
Consider the identity sin( − x) = cos x .
2
a) Verify the identity numerically, when x =
π
6
b) Verify the identity graphically.
c) Prove the identity algebraically
π π
π
3
- ) = sin =
2 6
3
2
π
3
Right side: cos =
6
2
π
b) The graphs of y1 = sin( - x) and y2 = cos x for -2π ≤ x ≤ 2π , appear to be
2
identical
a) Left side: sin(
c)
Left Side
π
- x)
2
π
π
= sin cos x - cos sin x
2
2
= (1)cos x - (0)sin x
= cos x
sin(
Right Side
cos x
Example 3:
2
3
and cos B = − and both ∠A and ∠B are in Quadrant 2, evaluate
3
5
cos( A − B) . (Use the sketches given to find the exact trig values.)
If sin A =
First calculate the other sides of the right angled triangles
2
4
1 =   + y2
5
2
2
16
4
y = 1 -  =
25
5
4
y=±
5
since y is in quadrant 2 it is positive.
2
2
2
4
1 =   + y2
5
2
2
16
4
y = 1 -  =
25
5
4
y=±
5
since y is in quadrant 2 it is positive.
2
2
Next use these values and the identity for cos(A - B) to find the exact value.
cos(A - B) = cos A cos B - sin A sin B
=(
- 5 -3
2 4
3 5 -8
)( ) - ( )( ) =
3
5
3 5
15
Example 4:
Rewrite cos
cos
π
12
π
12
as a difference identity and use the identity to find the exact value of
.
π π π π
To use exact values, try to involve angles such as 0, , , , , or π . In this case
6 4 3 2
π π π
π π π
= = or
12 3 4
12 4 6
cos
π
π π
= cos( - )
12
3 4
π
π
π
π
= cos cos + sin sin
3
4
3
4
1
1
3
1
= ( )(
)+ (
)(
)
2
2
2
2
=
1+ 3
2 2
Example 5:
1 − tan 2 x
1 + tan 2 x
a) Use the graph to make a conjecture about the period and the amplitude of f ( x) .
b) Make a conjecture stating what single trigonometric function f ( x) is equivalent to.
Graph this function.
c) Prove your conjecture algebraically.
Graph the function f ( x) =
The period of the graph is pi and the amplitude is one.
The graph is equivalent to the graph of y = cos 2 x .
sin 2 x
cos 2 x - sin 2 x cos 2 x - sin 2 x
1 - tan x
cos 2 x =
cos 2 x
cos 2 x
f(x) =
=
=
2
2
2
2
1
sin x cos x + sin x
1 + tan x
1+
2
2
cos 2 x
cos x
cos x
cos 2 x - sin 2 x
=
× cos 2 x = cos 2 x - sin 2 x = cos x cos x - sin x sin x
2
cos x
= cos(x + x)
= cos2x
2
1-