1. Factoring and Simplifying Trigonometric Expressions
Dr. Shildneck

2. How to Write Powers with Trig Functions
You might see a power in two different places when using trig functions. The position of the power means different things. 𝒔𝒊𝒏𝒙 𝟐 = 𝐬𝐢𝐧⁡(𝒙 𝟐 ) = 𝐬𝐢𝐧⁡(𝒙∙𝒙) versus 𝒔𝒊𝒏 𝟐 𝒙 = (𝒔𝒊𝒏𝒙) 𝟐 = (𝒔𝒊𝒏𝒙)(𝒔𝒊𝒏𝒙)

3. Example 1 - GCF 3 𝑐𝑠𝑐 3 𝑥−15 𝑐𝑠𝑐 2 𝑥 3 𝑢 3 −15 𝑢 2 = 3 𝑢 2 (𝑢−5)
Let 𝑢=𝑐𝑠𝑐𝑥
3 𝑐𝑠𝑐 3 𝑥−15 𝑐𝑠𝑐 2 𝑥
3 𝑢 3 −15 𝑢 2
= 3 𝑢 2 (𝑢−5)
= 3 𝑐𝑠𝑐 2 𝑥(𝑐𝑠𝑐𝑥−5)

4. Example 2 – Difference of Squares
Let 𝑢=𝑡𝑎𝑛𝑥
4 𝑡𝑎𝑛 2 𝑥−25
4 𝑢 2 −25
= (2𝑢−5) (2𝑢+5)
= (2𝑡𝑎𝑛𝑥−5) (2𝑡𝑎𝑛𝑥+5)

5. Example 3 – Quadratic Trinomial
Let 𝑢=𝑠𝑖𝑛𝑥
4 𝑠𝑖𝑛 2 𝑥+𝑠𝑖𝑛𝑥−3
4 𝑢 2 +𝑢−3
=(4𝑢−3)(𝑢+1)
=(4𝑠𝑖𝑛𝑥−3)(𝑠𝑖𝑛𝑥+1)

6. Example 4 – Sum/Difference of Cubes
Let 𝑢=𝑐𝑜𝑠𝑥
27 𝑐𝑜𝑠 3 𝑥+8
27 𝑢 3 +8
=(3𝑢+2)(9 𝑢 2 −6𝑢+4)
=(3𝑐𝑜𝑠𝑥+2)(9 𝑐𝑜𝑠 2 𝑥−6𝑐𝑜𝑠𝑥+4)

7. Example 4 – Rational Functions
2 𝑠𝑖𝑛 2 𝑥+3𝑠𝑖𝑛𝑥−2 3 𝑠𝑖𝑛 2 𝑥+10𝑠𝑖𝑛𝑥+8
2 𝑢 2 +3𝑢−2 3 𝑢 2 +10𝑢+8
Let 𝑢=𝑠𝑖𝑛𝑥
= (2𝑢−1)(𝑢+2) 3𝑢+4 (𝑢+2)
= (2𝑢−1) 3𝑢+4
= (2𝑠𝑖𝑛𝑥−1) 3𝑠𝑖𝑛𝑥+4