Sum and Difference Identities Part 1

Hey guys, it's Kevin ~ Today we went over sum and difference identities // we also got our midterm marks /:

The basic sum identities include:

• cos(α + ß) = cosα cosß - sinα sinß
• sin(α + ß) = sinα cosß + cosα sinß
• tan(α + ß) = tanα + tanß / 1 - tanα tanß

The basic difference identities are as follows:

• cos(α - ß) = cosα cosß + s
  inα sinß
• sin(α - ß) = sinα cosß - cosα sinß
• tan(α - ß) = tanα - tanß / 1 + tanα tanß

The reciprocal functions can also be found:

• csc(α + ß) = 1 / sin(α + ß)
• sec(α + ß) = 1 / cos(α + ß)
• cot(α + ß) = 1 / tan(α + ß) or cos(α + ß) / sin(α + ß)

We were asked to find exact values of functions, but it won't always be a special triangle. Let's use sin 7π/12. We can find two special triangles that can equal that when added or subtracted. Some examples that work are (4π/12 + 3π/12) or (10π/12 - 3π/12). It doesn't matter which set you use, you'll end up with the same answer. When solving these questions you need to be aware of the special triangles, SOH CAH TOA, and the CAST rule. You will be needing your formula sheet for this.

To solve sin 7π/12:

1. Find two special triangles that equal to 7π/12: sin(4π/12 + 3π/12) *note that 4π/12 is the α and 3π/12 is the ß
2. Convert: sin(π/3 + π/4)
3. Get the formula for sin(α + ß) from the formula sheet: sinα cosß + cosα sinß
4. Now plug in your information: sin π/3 • cos π/4 + cos π/3 • sin π/4
5. This is the part where you'll need to know your special triangles. If it helps, draw a diagram on the side. You need to know the opposite, adjacent, and hypotenuse sides of the triangles.


6. 
   Now use SOH CAH TOA and plug in the information: √3/2 • 1/√2 + 1/2 • 1/√2 *Just keep in mind the CAST rule and watch for which functions are negatives or positives.
7. Lastly, just finish up the calculations: √3/2√2 + 1/2√2 or √3 + 1/2√2
8. Leave your answer as √3 + 1/2√2, don't rationalize or anything.

*A little trick to solving reciprocal functions is to just solve the basic function and flip your answer. So if the question above was csc7π√12, your answer would be 2√2/√3 + 1.