32)+Verify+trig+IDs+with+sum,+difference,+double+angle+IDs

Use a sum or difference ID to calculate exact values of trig functions.

__ **Background:** __ This section was rather easy after doing a variety of problems. In the beginning, I wasn't sure about the purpose of the section, but after doing problems, I realized how it related back to the sections we had done before it. I personally liked this section because it wasn't a difficult and complicated geometrical type of section. These problems related more to an algebraic base as opposed to geometric, and I am an algebra type of person.

__ Explanation: __
Sum and difference identities show the sine, cosine, or tangent values of a given angle whether the angle is positive or negative. These identities can be angles that are not necessarily on the unit circle, being that they are either less than 0 or greater than 2 π. In order to solve these problems, think of factors which add or subtract to the number given, making sure that these angle measures are part of the unit circle. After finding these factors, choose the correct formula and substitute the numbers in the correct places, keeping in mind that α is the first number and β is the second. From here, you can evaluate the individual expressions for numbers which come from the unit circle. To finish the problem, solve until you get exact, rationalized numbers.

=__ Trig IDs you will need: __= __tanα+tanβ__ 1-tanαtanβ __tanα-tanβ__ 1+tanαtanβ
 * ** cosine sum identity **: cos(α+β) = cosβcosα-sinβsinα
 * ** cosine difference identity **: cos(α-β) = cosβcosα+sinβsinα
 * ** sine sum identity **: sin(α+β) = sinαcosβ+cosαsinβ
 * ** sine difference identity **: sin(α-β) = sinαcosβ-cosαsinβ
 * ** tan sum identity **: tan(α+β) =
 * ** tan difference identity **: tan(α-β) =

(A=α B=β)

Examples: 1) sin195 = sin(150+45) = sin150cos45+cos150sin45  = (1/2)( √ 2/2)+(- √ 3/2)( √ 2/2)    = √ 2/4 - √6 /4

2) sin75 = sin(30+45) =sin30cos45+cos30sin45   =(1/2)( √ 2/2) + ( √3 /2)( √ 2/2)    = √ 2/4 + √6 /4

3) tan(105) = tan(45+60) =__tan45+tan60__  1-tan45tan60   =__1+__ __ √3 __  1-1(√3)  =(1+√3)(1+√3)   =1+√3 + √3+3   =4+__2__√3

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Problem for the Portfolio: 1a) Use the tan sum identity to calculate the exact value of tan(75) 1b) Use the cosine difference identity to calculate the exact value of cos(30)

1c) Use the tan difference identity to calculate the exact value of tan(30)