Section 10.4: Sum and Difference Formulas
The following identities are provided without proof. You DO NOT need to memorize
them for the test, they will be provided. You only need to know how to use them.
Sum and Difference Formulas:
sin(u + v) = sin u cos v + cos u sin v
sin(u − v) = sin u cos v − cos u sin v
cos(u + v) = cos u cos v − sin u sin v
cos(u − v) = cos u cos v + sin u sin v
tan u + tan v
1 − tan u tan v
tan u − tan v
tan(u − v) =
1 + tan u tan v
tan(u + v) =
Note: These formulas are VERY SPECIFIC.
You CAN NOT distribute ANY
trigonometric function:
sin(u + v ) ≠ sin u + sin v
⎛ 7π π ⎞
Ex 1: Find the exact value of sin⎜
− ⎟ using the difference formula.
3⎠
⎝ 6
sin(u − v) = sin u cos v − cos u sin v
7π
π
and v =
In this case u =
v
3
⎛ 7π ⎞ ⎛ π ⎞
⎛ 7π ⎞ ⎛ π ⎞
⎛ 7π π ⎞
So sin⎜
− ⎟ = sin⎜ ⎟ cos⎜ ⎟ − cos⎜ ⎟ sin⎜ ⎟
3⎠
⎝ 6 ⎠ ⎝3⎠
⎝ 6 ⎠ ⎝3⎠
⎝ 6
To solve this we will draw triangles on the coordinate axes and find the values on the
right hand side of the expression:
Ex 2: Find the exact value of sin(195°).
To solve this we need to find two angles whose sum or difference is 195° and are easy
angles to solve. For example 195° = 220° – 30° or 195° = 135° + 60°
So we could solve this more than one way. We will use the sum formula.
sin(195°) = sin(135° + 60°) = sin(135°) cos(60°) + cos(135°) sin(60°)
Once again we need to draw triangles to solve the right hand side of the equation.
Ex 3: Write the expression as the sine, cosine, or tangent of an angle:
sin(140°) cos(50°) + cos(140°) sin(50°)
Here we will use the sum formula for sine:
sin(140°) cos(50°) + cos(140°) sin(50°) = sin (140° +50°) = sin(190°)
⎛ π ⎞ ⎛ 3π ⎞
⎛ π ⎞ ⎛ 3π ⎞
Ex 4: Find the exact value cos⎜ ⎟ cos⎜ ⎟ − sin⎜ ⎟ sin⎜ ⎟
⎝ 16 ⎠ ⎝ 16 ⎠
⎝ 16 ⎠ ⎝ 16 ⎠
The next example is very similar to a problem on the test:
Ex 5: Suppose sin u = 5/13 and u is in quadrant I and cos v = -3/5 and v is in quadrant III.
Find cos (u – v)
We need to draw one reference triangle for each angle and then apply the formula:
cos(u − v) = cos u cos v + sin u sin v