1. Sum and Difference Identities for Cosine

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

3. 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."

4. 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.

5. The sum of all of the angles in a triangle always is 180°   a b c 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. adjacent to  opposite  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°

6. Looking at the names of the other trig functions can you guess which ones are cofunctions of each other?   a b c Let's see if this is right. Does sec  = csc  ? adjacent to  opposite  adjacent to  opposite  secant and cosecanttangent and cotangent hypotenuse over adjacent hypotenuse over opposite This whole idea of the relationship between cofunctions can be stated as: Cofunctions of complementary angles are equal.

7. 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? The sum of complementary angles in radians is since 90° is the same as Basically any trig function then equals 90° minus or minus its cofunction.

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