PRE-CALCULUS /TRIGONOMETRY
Name _______________________
Objective:
a. Verify trigonometric identities with factoring and simplifying rational expressions.
Factor - Review
Algebra
Trigonometry
1) 9x2 – 16
1) sin2x-cos2x
2) 36rs2 – 48r2s
2) 3sinx cosx + 6cosx
3) m2 – 6m + 9
3) cot2x – 8cotx + 16
4) y2 – 5y – 84
4) cos2x + 2cosx – 35
5) 6c2 + 13c – 5
5) 8csc2x – 2cscx – 1
6) 6p2 – 5pj – 4j2
6) sin2x – 2sinx cosx + cos2x
Simplifying Rational Expressions (be sure there are no negative exponents)
Algebra
1)
2)
3)
4)
3𝑥 2 𝑦
1)
12𝑦 2 𝑥
4𝑥 3
3𝑦 2
∙( )
3𝑦 2
2𝑥
𝑥 2 −4
2)
3)
𝑥+2
4𝑥 2 −9 𝑦 2 −1
3𝑦+3
Trigonometry
∙
6𝑥−9
4)
𝑠𝑖𝑛2 𝑥𝑐𝑜𝑠𝑥
𝑐𝑜𝑠 2 𝑥𝑠𝑖𝑛𝑥
4𝑐𝑜𝑠 3 𝑥
𝑠𝑖𝑛3 𝑥
∙(
𝑠𝑖𝑛𝑥
)
4𝑐𝑜𝑠𝑥
2
𝑠𝑒𝑐 2 𝑥−9
𝑠𝑒𝑐𝑥−3
9𝑡𝑎𝑛2 𝑥−25
2𝑡𝑎𝑛𝑥−2
∙
𝑡𝑎𝑛2 𝑥−1
6𝑡𝑎𝑛𝑥−10
Examples - Factor
1) 3cot2x + 6cotx + 3
2) 2cos2x + cosx – 3
3) 2sin2y + 9siny + 4
4) 3sin2y – 7 siny – 20
Examples – Simplify
1)
3)
30𝑠𝑖𝑛3 𝑥𝑐𝑜𝑠𝑥
2)
6𝑐𝑜𝑠 2 𝑥𝑠𝑖𝑛𝑥
𝑠𝑒𝑐 4 𝑥−𝑡𝑎𝑛4 𝑥
4)
𝑠𝑒𝑐 2 𝑥−𝑡𝑎𝑛2 𝑥
Examples – Verify
1)
2)
3)
𝑐𝑜𝑠 2 𝑥+3𝑐𝑜𝑠𝑥+2
𝑐𝑜𝑠𝑥+1
𝑠𝑖𝑛4 𝑥−𝑐𝑜𝑠 4 𝑥
𝑠𝑖𝑛2 𝑥−𝑐𝑜𝑠 2 𝑥
= 𝑐𝑜𝑠𝑥 + 2
= 𝑠𝑖𝑛2 𝑥 + 𝑐𝑜𝑠 2 𝑥
𝑐𝑜𝑡𝑥𝑠𝑖𝑛2 𝑥+𝑐𝑜𝑡𝑥𝑐𝑜𝑠 2 𝑥
𝑐𝑜𝑠 2 𝑥𝑐𝑠𝑐𝑥𝑡𝑎𝑛𝑥𝑠𝑒𝑐𝑥
= 𝑐𝑜𝑡𝑥
𝑠𝑖𝑛2 𝑥−9
2𝑐𝑜𝑠𝑥+1
∙
10𝑐𝑜𝑠𝑥+5
3𝑠𝑖𝑛𝑥+9
2𝑐𝑜𝑠 2 𝑥−𝑐𝑜𝑠𝑥−1
𝑐𝑜𝑠𝑥−1