5-4: Sum and Difference Formulas
A proof of the sine sum formula can be found at www.khanacademy.org ->math->trigonometry->proof of
sin(a+b) = sin(a)cos(b)+sin(b)cos(a)
Sum and Difference Formulas
Cosine:
cos (A + B) = cos A cos B – sin A sin B
cos(A – B) = cos A cos B + sin A sin B
Sine:
sin(A + B) = sin A cos B + cos A sin B
sin(A – B) = sin A cos B – cos A sin B
Tangent:
tan( A  B) 
tan A  tanB
1  tan A tan B
tan( A  B) 
tan A  tanB
1  tan A tan B
The sum and difference formulas can be used to find the exact value of an angle whose sum or difference
comes from two “special angles” – angles that have reference angles of 30°, 45°, or 60°.
Example:
Find the exact value of cos 15°. 15° is not a “special angle”, but 60° – 45° = 15° and 60° and 45° are special
angles. Since I subtracted to get 15, I use the difference formula. A = 60° and B = 45°.
cos 15° = cos(60° – 45°) = cos 60° cos 45° + sin 60° sin 45°
1
2
3
2
= ( )( )  ( )( )
2 2
2
2
=
2
6

4
4
=
2 6
4
45 – 30 = 15, 135 – 120 = 15, -30 + 45 = 15, 240 – 225 = 15, -315 + 330 = 15 by using any of these
sums or differences, we would have obtained the same answer. The angles you pick just have to have a
reference angle of 30, 45 or 60. Therefore, 50 – 35 = 15, but those are two angles that cannot be used.
1. Just figure out some angles that add or subtract to give us the given angle. Once you have that figured
out, it is just a matter of plugging into a formula. List 2 sets of angles for each given angle.
ex: 75°
105°
195°
345°
11
12
30 + 45
210 – 135
Use the formulas from the front and the sets of angles from above to find each.
2. Find the exact value of sin 195°.
3. Find the exact value of tan 345°.
Find the exact value of each given expression.
4. cos 130°cos 40° + sin 130° sin 40°
This looks like the right side of the cosine difference identity, right? A = 130° and B = 40°.
Therefore, we have cos (130° − 40°) = cos 90° = 0
5. cos 170° cos 190° − sin 170° sin 190°
Which sum or difference identity does this look like?
6. sin 220° cos 40° − cos 220°sin 40°
7.
tan110  tan 50
1  tan110 tan 50
Find all solutions of the equation in the interval *0, 2π).
8. sin( x 

6
)  sin( x 

6
)
1
2
9. cos( x 

4
)  cos( x 

4
) 1
Verify the identity.
10.
sin(

2
 x)  cos x
11. cos(
5
2
 x)  
(cos x  sin x)
4
2
12. cos( x  y)  cos( x  y)  2 cos x cos y
Find the exact value of each trig functions given that sin α = 5/13 and cos β = -3/5. Both α and β are in
quadrant II.
13. sin (α + β)
15. sec (β – α)
14. tan (α – β)