# Trigonometry - Addition & SubtractionFormulas

## Trigonometry - Addition and Subtraction Formulas - How it Works - Video

### Unit Circle

Unit Circle:

The unit circle is vital to answer any exact answer. Do you have to memorize every bit? That is up to. It would be faster if you had it encoded in your brain, but you can memorized just the first quadrant and use symmetry to fill out the unit circle.

sin(u + v) = sin u * cos v + cos u * sin v

cos(u + v) = cos u * cos v - sin u * sin v

tan(u + v) = [tan u + tan v] / [ 1 - tan u * tan v]

Subtraction Formulas:

sin(u - v) = sin u * cos v - cos u * sin v

cos(u - v) = cos u * cos v + sin u * sin v

tan(u - v) = [tan u - tan v] / [ 1 + tan u * tan v]

The way that I remember the signs if there is not a formula chart sheet is that cosign is that only one that flips the sign.

Tangent has the same on top and different on bottom, and that makes sense since tangent is sine over cosign. If we take a look at the signs for addition, sine is positive and cosine is negative, and tangent is positive over negative.

If we take a look at the signs for subtraction, sine is negative and cosine is positive, and tangent is negative over positive.

### Example 1 Part 1

Example 1 Part 1:

Our first step is to separate 7π /12 in cos(7π /12). We can either use addition or subtraction formula for cosine. Let's start with the additions formula.

7π /12 = 1π /12 + 6π /12 => π /12 + π /2

7π /12 = 2π /12 + 5π /12 => π /6 + 5π /12

7π /12 = 3π /12 + 4π /12 => π /4 + π /3

7π /12 = 4π /12 + 3π /12 => π /3 + π /4

7π /12 = 5π /12 + 2π /12 => 5π /12 + π /6

7π /12 = 6π /12 + 1π /12 => π /2 + π /12

We start with 1π /12 + 6π /12 and simplify. Our result is π /12 + π /2. Only one of those numbers is on the unit circle, so we have to try again, and that's okay. It will help us the next time we see a similar number.

Lucky for us, two different pairings work, but once you find one, you are good to go. We are going to use 7π /12 = 3π /12 + 4π /12 => π /4 + π /3 our third guess to find the exact value using the addition formula.

### Example 1 Part 2

Example 1 Part 2:

Once we have two numbers we substitute the numbers into the formula, cos(u + v) = cos u * cos v - sin u * sin v.

So we get cos(π /4 + π /3) = cos π /4 * cos π /3 - sin π /4 * sin π /3, where π /4 = u and π /3 = v.

=> cos(π /4 + π /3) = cos π /4 * cos π /3 - sin π /4 * sin π /3

=> sqrt(2) / 2 * 1/2 - sqrt(2) / 2 * sqrt(3) / 2

=> sqrt(2) / 4 - sqrt(6) / 4

=> [sqrt(2) - sqrt(6)] / 4

Substituted u and v.

Use the unit chart to find the corresponding x and y values.

Multiplied numerators and multiplied denominators.

Combined fractions.

So our exact is [sqrt(2) - sqrt(6)] / 4.

### Example 2 Part 1

Example 2 Part 1:

Our first step is to separate 7π /12 in cos(7π /12). Since we used addition already, let's use the subtraction formula for cosine.

7π /12 = 12π /12 - 5π /12 => π + 5π /12

7π /12 = 11π /12 - 4π /12 => 11π/12 + π /3

7π /12 = 10π /12 - 3π /12 => 5π/6 + π /4

7π /12 = 9π /12 - 2π /12 => 3π/4 + π /6

7π /12 = 8π /12 - 1π /12 => 2π/3 + π /12

We start with 12π /12 - 5π /12 and simplify. Our result is π - 5π /12. Only one of those numbers is on the unit circle, so we have to try again, and that's okay. It will help us the next time we see a similar number.

Lucky for us, two different pairings work, but once you find one, you are good to go. We are going to use 7π /12 = 10π /12 - 3π /12 => 5π/6 + π /4 our third guess to find the exact value using the subtraction formula.

### Example 2 Part 2

Example 2 Part 2:

Once we have two numbers we substitute the numbers into the formula, cos(u - v) = cos u * cos v + sin u * sin v.

So we get cos(5π /6 - π /4) = cos 5π /6 * cos π /4 - sin 5π /6 * sin π /4, where 5π /6 = u and π /4 = v.

=> cos(5π /6 - π /4) = cos 5π /6 * cos π /4 - sin 5π /6 * sin π /4

=> -sqrt(3) / 2 * sqrt(2) / 2- 1 / 2 * sqrt(2) / 2

=> -sqrt(6) / 4 + sqrt(2) / 4

=> [-sqrt(6) + sqrt(2)] / 4

Substituted u and v.

Use the unit chart to find the corresponding x and y values.

Multiplied numerators and multiplied denominators.

Combined fractions.

So our exact is [-sqrt(6) + sqrt(2)] / 4.

Sine and tangent work the same way.