Factoring and Simplifying Trigonometric Expressions Dr. Shildneck

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)

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)

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)

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)

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)

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