Sum and Difference

Identities for Cosine

Consider the equation

cos cos cos Is this an identity? Remember an identity means the equation is true for every value of the variable for which it is defined.

Let’s try = 30° and β = 45°

?

cos 30 45 cos30 cos 45

cos 75 0.2588 cos30 cos 45 1.573

So cos 30 45 cos30 cos 45

This is NOT an identity and DOES NOT WORK for all values!!!

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 would like an identity to express this in terms of products and sums of sines and cosines. The proof of this identity is on Page 185-186 in your book. The identities 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.

The sum of all of the angles in a triangle always is 180°

a

bc

What is the sum of + ?

Since we have a 90° angle, the sum of the other two angles must also be 90° (since the sum of all three is 180°).

Two angles whose sum is 90° are called

complementary angles.

?sin isWhat c

a

?os isWhat cc

a

adja

cen

t to

op

pos

ite

adjacent to opposite

Since and are complementary angles and

sin = cos , sine and cosine are called

cofunctions.

This is where we get the name cosine, a cofunction of sine.

90°

Looking at the names of the other trig functions can you guess which ones are cofunctions of each other?

a

bc

Let's see if this is right. Does sec = csc ?

cscsec b

c

adja

cen

t to

op

pos

ite

adjacent to opposite

secant and cosecant tangent and cotangent

hypotenuse over adjacenthypotenuse over

opposite

This whole idea of the relationship between cofunctions can be stated as:

Cofunctions of complementary angles are equal.

Cofunctions of complementary angles are equal.

cos 27°Using the theorem above, what trig function of what angle does this equal?

= sin(90° - 27°) = sin 63°

Let's try one in radians. What trig functions of what angle does this equal?

8tan

82cot

The sum of complementary angles in radians is since 90° is the same as 2

2

8

3cot

Basically any trig function then equals 90° minus or minus its cofunction.

2

Cofunction Identities

sin2

u

cosu cos

2

u

sinu

tan2

u

cotu cot

2

u

tanu

sec2

u

cscu csc

2

u

secu

54sin

36sin

We can't use fundamental identities if the trig functions are of different angles.

Use the cofunction theorem to change the denominator to its cofunction

36cos

36sin

Now that the angles are the same we can use a trig identity to simplify.

36tan