Sum and Difference Formulas

cos cos cos sin sin

cos cos cos sin sin

Often you will have the cosine of the sum or difference of two angles. We are going to use formulas for this to express in terms of products and sums of sines and cosines. The formulas are:

You will need to know these so say them in your head when you write them like this, "The cosine of the sum of 2 angles is cosine of the first, cosine of the second minus sine of the first sine of the second."

.105cos of eexact valu theFind

cos cos105 60 45

cos cos sin sin60 45 60 45

2

2

2

3

2

2

2

1

24

64

2 64

Since it says exact we want to use values we know from our unit circle. 105° is not one there but can we take the sum or difference of two angles from unit circle and get 105° ?

We can use the sum formula and get cosine of the first, cosine of the second minus sine of the first, sine of the second.

sin sin cos cos sin

sin sin cos cos sin

We have a formula for sum or difference of angles for the sine function as well. The proof of this one is on page 618 in your book. The formulas are:

Sum of angles for sine is, "Sine of the first, cosine of the second plus cosine of the first sine of the second." You can remember that difference is the same formula but with a negative sign.

.12

sin of eexact valu theFind

34sin

12sin

sin cos sin cos 4 3 3 4

2

2

2

3

2

1

2

2

24

64

2 64

A little harder because of radians but ask, "What angles on the unit circle can I add or subtract to get negative pi over 12?" hint: 12 is the common denominator between 3 and 4.

You will need to know these formulas so let's study them a minute to see the best way to memorize them.

cos cos cos sin sin

cos cos cos sin sin

sin sin cos cos sin

sin sin cos cos sin

cos has same trig functions in first term and in last term, but opposite signs between terms.

opposite

sin has opposite trig functions in each term but same signs between terms.

same

There are also sum and difference formulas for tangent that come from taking the formulas for sine and dividing them by formulas for cosine and simplifying (since tangent is sine over cosine).

tantan tan

tan tan

1

tantan tan

tan tan

1

Combined Sum and Difference Formulas

sinsincoscoscos

sincoscossinsin

tantan1

tantantan

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au