Sum and Difference Formulas 12.3 Sum and Difference Formulas

Unit Circle Review

What if you are looking for an angle you do not know, how would you even try to solve it? Try breaking it into a couple of angles you do know.

Since it says exact we want to use values we know from our unit circle 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° ?

Often you will have the cosine of the sum or difference of two angles 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 sum or difference of angles for the cosine function 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."

Let’s look back and solve that equation that we started with: 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 or difference of angles for the sine function 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.

hint: 12 is the common denominator between 3 and 4. Find the exact value of 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 has same trig functions in first term and in last term, but opposite signs between terms.

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

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).

Combined Sum and Difference Formulas

Let’s try some: