Sum-Difference Identities

SUM-DIFFERENCE IDENTITIES:

​​

Sum, Difference, & Double Angle Identities



Angle Sum and Difference Theorem: The following identities are true for all values for which they are defined:
external image anglesum01.gif

cos(x - y) = cos(x)cos(y) + sin(x)sin(y)

cos(x + y)=cos(x)cos(y) -- sin(x)sin(y)

sin(x - y)=sin(x)cos(y) -- cos(x)sin(y)

sin(x + y)=sin(x)cos(y) + cos(x)sin(y)

gsdfdgdfg tan(x)+tan(y)
tan(x+y)= ----------------------
as asdf 1-tan(x)tan(y)

sdfggsdfg tan(x)-tan(y)
tan(x-y)= ----------------------
sssssss 1+tan(x)tan(y)



Although the equations look long, it is really just simply a matter of plugging in the numbers.

Don't get too worked up about the x's and y's, you could even substitute them for A and B if you'd like.
As long as x sine and x cosine are from the same number and the y sine and y cosine are from the
same, you should be good to go.

Here is an example of proving sum and difference identities.

cos(x-y) + cos(x+y) = 2cos(x)cos(y)
cos(x)cos(y) + sin(x)sin(y) + cos(x)cos(y) - sin(x)sin(y) = 2cos(x)cos(y)
2cos(x)cos(y) = 2cos(x)cos(y)


--->> Here is a "cheat sheet" for when you start using Tangent from the Unit Circle in the formulas state above.
Don't forget!!! To find tangent, the formula is tan=y/x, (meaning tangent = y over x).



Degrees
Radians
Tangent Measure
30°
π/6
(√3)/3
45°
π/4
1
60°
π /3
(√3)
90°
π/2
Undefined
120°
2π/3
(-√3)
135°
3π/4
-1
150°
5π/6
(-√3)/3
180°
π
0
210°
7π/6
(√3)/3
225°
5π/4
1
240°
4π/3
(√3)
270°
3π/2
Undefined
300°
5π/3
(-√3)
315°
7π/4
-1
330°
11π/6
(-√3)/3
360°
2π
0



















Unit CircleUNIT CIRCLE

external image unit_circle.gif


Proof Practice


Proof: Let P be the point with coordinates external image trigdef02.gif. Measured counterclockwise from point P, let Q be the point whose arc length is A, let R be the point whose arc length is external image anglesum02.gif, and let S be the point whose arc length is external image anglesum03.gif. Then external image anglesum04.gif are the coordinates of point Q, external image anglesum05.gif are the coordinates of point R, and external image anglesum06.gif are the coordinates of point S.
external image anglesum07.gif
Note that the lengths of segments PR and QS are equal. Using the distance formula, we get:
external image anglesum08.gif
Through the use of the symmetric and Trigonometric Identities , this simplifies to become the angle sum formula for the cosine.

The formulas can also be derived using triangles. Although we refer to the following derivation as a proof, in fact the values of angles A and B allowed by the derivation are quite limited, and a more general proof is actually required











.

Angle Difference Formula

  • sin(A − B) = sinAcosB − cosAsinB
  • cos(A − B) = cosAcosB + sinAsinB
These formula allow you to express the exact value of trigonometric expressions that you could not otherwise express. Consider the cos(15°). Unlike the cos(60°) which can be expressed as ½, the cos(15°) cannot simply be represented as a rational expression. However, the angle difference formula allows you to represent the exact value of this function.
Picture of angle difference formula
Picture of angle difference formula

NOW YOU TRY IT! Try these sample problems. Just highlight the text to see the answer!

cos15=?

  • cos15=cos(60-45)
  • We know that cos(x-y)=cosxcosy+sinxsiny, so
  • cos(60-45)=cos60cos45+sin60sin45
  • cos(60-45)=(1/2)(√2/2)+(√3/2)(√2/2)
  • cos(60-45)=((√2)+√6)/2

Angle Sum Formula

  • sin(A+B) = sinAcosB + cosAsinB
  • cos(A+B) = cosAcosB − sinAsinB
These formulas allow you to express the exact value of trigonometric expressions that you could not otherwise express. Consider the sin(105°). Unlike the sin(30) which can be expressed as ½, the sin(105) cannot simply be represented as a rational expression. However, the angle sum formula allows you to represent the exact value of this function
Picture of angle sum formula
Picture of angle sum formula


So to wrap things up: