Trig Identities

Question 1

sin²(x)

Answer 1

1 - cos²(x)

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

Question 2

cos²(x)

Answer 2

1 - sin²(x)

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

Question 3

sec²(x)

Answer 3

1 + tan²(x)

Question 4

tan²(x)

Answer 4

sec²(x) - 1

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

Question 5

csc²(x)

Answer 5

1 + cot²(x)

Question 6

cot²(x)

Answer 6

csc²(x) - 1

Question 7

Pythagorean Identities

Answer 7

sin²(x) + cos²(x) = 1
1 + tan²(x) = sec²(x)
1 + cot²(x) = csc²(x)

Question 8

sin(π/2 - x)

Answer 8

cos(x)

Question 9

csc(π/2 - x)

Answer 9

sec(x)

Question 10

sec(π/2 - x)

Answer 10

csc(x)

Question 11

cos(π/2 - x)

Answer 11

sin(x)

Question 12

tan(π/2 - x)

Answer 12

cot(x)

Question 13

cot(π/2 - x)

Answer 13

tan(x)

Question 14

Cofunction Identities

Answer 14

sin(π/2 - x) = cos(x)
cos(π/2 - x) = sin(x)
tan(π/2 - x) = cot(x)
csc(π/2 - x) = sec(x)
sec(π/2 - x) = csc(x)
cot(π/2 - x) = tan(x)

Question 15

sin(a + b)

Answer 15

sin(a)cos(b) + sin(b)cos(a)

Question 16

sin(a - b)

Answer 16

sin(a)cos(b) - sin(b)cos(a)

Question 17

cos(a + b)

Answer 17

cos(a)cos(b) - sin(a)sin(b)

Question 18

cos(a - b)

Answer 18

cos(a)cos(b) + sin(a)sin(b)

Question 19

tan(a + b)

Answer 19

(tan(a) + tan(b))/(1 - tan(a)tan(b))

Question 20

tan(a - b)

Answer 20

(tan(a) - tan(b))/(1 + tan(a)tan(b))

Question 21

Sum and Difference Formulas

Answer 21

sin(a + b) = sin(a)cos(b) + sin(b)cos(a)
sin(a - b) = sin(a)cos(b) - sin(b)cos(a)
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b))
tan(a - b) = (tan(a) - tan(b))/(1 + tan(a)tan(b))

Question 22

sin(2x)

Answer 22

2sin(x)cos(x)

Question 23

cos(2x)

Answer 23

cos²(x) - sin²(x)
2cos²(x) - 1
1 - 2sin²(x)

Question 24

tan(2x)

Answer 24

(2tan(x))/(1 - tan²x)

Question 25

Double Angle Formulas

Answer 25

``` sin(2x) = 2sin(x)cos(x) cos(2x) = cos²(x) - sin²(x) cos(2x) = 2cos²(x) - 1 cos(2x) = 1 - 2sin²(x) tan(2x) = (2tan(x))/(1 - tan²x) ```

Question 26

sin(a) + sin(b)

Answer 26

2sin((a+b)/2)cos((a-b)/2)

Question 27

sin(a) - sin(b)

Answer 27

2cos((a+b)/2)sin((a-b)/2)

Question 28

cos(a) + cos(b)

Answer 28

2cos((a+b)/2)cos((a-b)/2)

Question 29

cos(a) - cos(b)

Answer 29

-2sin((a+b)/2)sin((a-b)/2)

Question 30

Sum-to-Product Formulas

Answer 30

sin(a) + sin(b) = 2sin((a+b)/2)cos((a-b)/2) sin(a) - sin(b) = 2cos((a+b)/2)sin((a-b)/2) cos(a) + cos(b) = 2cos((a+b)/2)cos((a-b)/2) cos(a) - cos(b) = -2sin((a+b)/2)sin((a-b)/2)

Question 31

sin(a)sin(b)

Answer 31

1/2[cos(a - b) - cos(a + b)]

Question 32

cos(a)cos(b)

Answer 32

1/2[cos(a - b) + cos(a + b)]

Question 33

sin(a)cos(b)

Answer 33

1/2[sin(a + b) + sin(a - b)]

Question 34

cos(a)sin(b)

Answer 34

1/2[sin(a + b) - sin(a - b)]

Question 35

Product to Sum Formulas

Answer 35

``` sin(a)sin(b) = 1/2[cos(a - b) - cos(a + b)] cos(a)cos(b) = 1/2[cos(a - b) + cos(a + b)] sin(a)cos(b) = 1/2[sin(a + b) + sin(a - b)] cos(a)sin(b) = 1/2[sin(a + b) - sin(a - b)] ```

Question 36

Circular Function Definitions

Answer 36

``` 0 < Ɵ < π/2 sin(Ɵ) = y/r cos(Ɵ) = x/r tan(Ɵ) = y/x csc(Ɵ) = r/y sec(Ɵ) = r/x cot(Ɵ) = x/y ```

Question 37

Right Triangle Definitions

Answer 37

``` sin(Ɵ) = opp/hyp cos(Ɵ) = adj/hyp tan(Ɵ) = opp/adj csc(Ɵ) = hyp/opp sec(Ɵ) = hyp/adj cot(Ɵ) = adj/opp ```

Question 38

√(a² - b²x²)

Answer 38

x = (a/b)sin(Ɵ) | 1 - sin²(Ɵ) = cos²(Ɵ)

Question 39

√(a² + b²x²)

Answer 39

x = (a/b)tan(Ɵ) | 1 + tan²(Ɵ) = sec²(Ɵ)

Question 40

√(b²x² - a²)

Answer 40

x = (a/b)sec(Ɵ) | sec²(Ɵ) -1 = tan²(Ɵ)

Question 41

d/dx(sin(x))

Answer 41

cos(x)

Question 42

d/dx(cos(x))

Answer 42

-sin(x)

Question 43

d/dx(tan(x))

Answer 43

sec²(x)

Question 44

d/dx(cot(x))

Answer 44

-csc²(x)

Question 45

d/dx(sec(x))

Answer 45

sec(x)tan(x)

Question 46

d/dx(csc(x))

Answer 46

-csc(x)cot(x)

Question 47

∫cos(x)dx

Answer 47

sin(x) + C

Question 48

∫sin(x)dx

Answer 48

-cos(x) + C

Question 49

∫sec²(x)dx

Answer 49

tan(x) + C

Question 50

∫csc²(x)dx

Answer 50

-cot(x) + C

Question 51

∫sec(x)tan(x)dx

Answer 51

sec(x) + C

Question 52

∫csc(x)cot(x)dx

Answer 52

-csc(x) + C

Question 53

∫tan(x)dx

Answer 53

-ln|cos(x)| + C

Question 54

∫cot(x)dx

Answer 54

ln|sin(x)| + C

Question 55

∫sec(x)dx

Answer 55

ln|sec(x) + tan(x)| + C

Question 56

∫csc(x)dx

Answer 56

ln|csc(x) - cot(x)| + C

Question 57

Half Angle Formulas

Answer 57

``` sin²(x) = [1 - cos(2x)]/2 cos²(x) = [1 + cos(2x)]/2 tan²(x) = [1 - cos(2x)]/[1 + cos(2x)] ``` ``` sin(x/2) = ± √[(1 - cos(x))/2)] cos(x/2) = ± √[1 + cos(x)/2)] tan(x/2) = ± √[(1 - cos(x))/(1 + cos(x))] tan(x/2) = (1 - cos(x))/(sin(x)) tan(x/2) = (sin(x))/(1 + cos(x)) ```

Question 58

sin²(x)cos²(x)

Answer 58

[1/2sin(2x)]²