6.2 Trigonometric Symbols and Substitution

Question 1

sin²(x) (even powers)

Answer 1

(1/2)(1-cos(2x))

Question 2

cos²(x) (even powers)

Answer 2

(1/2)(1+cos(2x))

Question 3

sin²(x) (odd powers)

Answer 3

(1 - cos²(x))

Question 4

cos²(x) (odd powers)

Answer 4

(1 - sin²(x))

Question 5

tan²(x) (even or odd powers)

Answer 5

sec²(x) - 1

Question 6

sec²(x) (even or odd powers)

Answer 6

tan²(x) + 1

Question 7

√a² + x²

Answer 7

x = a*tan(θ)

Question 8

√a² - x²

Answer 8

x = a*sin(θ)

Question 9

√x² - a²

Answer 9

x = a*sec(θ)

Question 10

√a² - a²sin²(θ)

Answer 10

a*cos(θ)

Question 11

√a² - a²tan²(θ)

Answer 11

a*sec(θ)

Question 12

√a² - a²sec²(θ)

Answer 12

a*tan(θ)

Question 13

range of a*sin(θ)

Answer 13

-(π/2)

Question 14

range of a*tan(θ)

Answer 14

-(π/2)

Question 15

range of a*sec(θ)

Answer 15

0

Question 16

Integrate even powers of sin or cos

Answer 16

1. Use the double-angle identity
2. Pull out constants
3. Integrate (use u-sub if necessary)

Question 17

Integrate odd powers of sin or cos

Answer 17

1. Split into even and odd power multiplied
2. Use the additive identity on even power
3. Integrate w/u-sub

Question 18

Integrate even powers of tan

Answer 18

1. Use additive Identity

2. Integrate wlu-sub

Question 19

Integrate odd powers of tan

Answer 19

1. Split into even and odd power multiplied
2. Use the additive identity on even power
3. Integrate w/u-sub

Question 20

Using trigonometric substitution

Answer 20

1. solve for x w/respect to θ
2. solve for dθ
3. substitute in for x and dx
4. simplify expressions using identities
5. integrate w/respect to θ
6. create reference triangle and solve for trig functions w/respect to x if needed
7. re-sub x in for θ