Trigonometric Identities

Question 1

which function does sin have a reciprocal relationship with?

Answer 1

csc

Question 2

which function does cos have a reciprocal relationship with?

Answer 2

sec

Question 3

which function does tan have a reciprocal relationship with?

Answer 3

cot

Question 4

which function does csc have a reciprocal relationship with?

Answer 4

sin

Question 5

which function does sec have a reciprocal relationship with?

Answer 5

cos

Question 6

which function does cot have a reciprocal relationship with?

Answer 6

tan

Question 7

power reduction of sin^2(u)

Answer 7

1/2(1-cos(2u))

Question 8

power reduction of cos^2(u)

Answer 8

1/2(1+cos(2u))

Question 9

power reduction of tan^2(u)

Answer 9

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

Question 10

double angle identity: sin(2u)

Answer 10

2sinucosu

Question 11

double angle identity: cos(2u)

Answer 11

cos^2(u) - sin^2(u)

Question 12

pythagorean identity: tan^2(u)

Answer 12

sec^2(u) - 1

Question 13

pythagorean identity: sec^2(u)

Answer 13

tan^2(u) + 1

Question 14

substitution: sqrt(a^2 + u^2)

Answer 14

u = (a)(tan(theta))

Question 15

substitution: sqrt(a^2 - u^2)

Answer 15

u = (a)(sin(theta))

Question 16

substitution: sqrt(u^2 - a^2)

Answer 16

u = (a)(sec(theta))

Question 17

arc length s of curve f(x)

Answer 17

s = definite integral from a to b of sqrt(1 + (f’(x))^2) dx

Question 18

Trap (n) = , also delta x =

Answer 18

(1/2)(delta x)[f(x,0) + 2f(x,1) + 2f(x,2) + … … + 2f(x,n-1) + f(x,n)]

delta x = (upper bound of integral - lower bound of integral)/(number of sub intervals ie terms in series)

Question 19

Mid (n) = , also delta x =

Answer 19

(delta x)[f(x,1) + f(x,2) … … + f(x,n)]

delta x = (upper bound of integral - lower bound of integral)/(number of sub intervals ie terms in series)

Question 20

Simp (n) = , also delta x =

Answer 20

(1/3)(delta x)[f(x,0) + 4f(x,1) + 2f(x,2) + 4f(x,3) + 2f(x,4) … … 4f(x,n-1) + f(x,n)]

n must be an even number.

Question 21

separable differential equation

Answer 21

dy/dx = (f(y))(g(x))

when derivative is isolated the other side of the equation can be factored so that one factor is a function of only y and the other factor is a function of only x.

Question 22

define implicit form of a differential equation

Answer 22

implicit form is not solved for y in terms of x (y is not completely isolated on one side)

Question 23

define explicit form of a differential equation

Answer 23

explicit form is solved for y in terms of x (y is completely isolated on one side)

Question 24

Integral Identity: integral of u^n du , when n does not equal 1

Answer 24

u^(n+1) / n+1 + C

Question 25

Integral Identity: integral of u^-1 du

Answer 25

ln|u| + C

Question 26

Integral Identity: integral of e^u

Answer 26

e^u + C

Question 27

Integral Identity: integral of a^u , when a does not equal 1

Answer 27

(1 / (lna) ) (a^u) + C

Question 28

Trig Integral Identity: integral of cos(u)

Answer 28

sin(u) + C

Question 29

Trig Integral Identity: integral of sin(u)

Answer 29

-cos(u) + C

Question 30

Trig Integral Identity: integral of (sec u)(tan u) du "a sea can tan if a sea can can."

Answer 30

sec(u) + C

Question 31

Trig Integral Identity: integral of sec^2(u) du "A sea can square its t...."

Answer 31

tan(u) + C "A sea can square its toes"

Question 32

Trig Integral Identity: integral of (csc u)(cot u) du "a cosy cot can if a cosy can can't."

Answer 32

-csc(u) + C

Question 33

Trig Integral Identity: integral of csc^2(u) du "a cozy can square without a cot there."

Answer 33

-cot(u) + C

Question 34

Trig Integral Identity: integral of tan(u) du "that tan has no lines cuz!"

Answer 34

-ln| cos(u) | + C

Question 35

Trig Integral Identity: integral of cot(u) du "that cot lines its signs!"

Answer 35

ln| sin(u) | + C

Question 36

Trig Integral Identity: integral of sec(u) du "A sea can always lines it's sea cans with tans."

Answer 36

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

Question 37

Trig Integral Identity: integral of csc(u) du "A cosy can alone can't line it's coats with cosy cans."

Answer 37

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

Question 38

Trig Integral Identity: integral of 1 / sqrt(a^2 - u^2) du

Answer 38

sin^-1 (u/a) + C

Question 39

Trig Integral Identity: integral of 1 / sqrt(a^2 + u^2) du

Answer 39

(1/a)(tan^-1 (u/a)) + C

Question 40

Integral Identity: integral of ln(u)

Answer 40

(x)(ln(x)) - x + C

Question 41

Integral Identity: integral of (u)(dv) , ie two statements multiplied

Answer 41

(u)(v) - integral of (v)(du)

Question 42

Trig Integral Identity: integral of 1 / ((u)(sqrt(u^2 - a^2) ) )

Answer 42

(1/a)(sec^-1( | u/a | ) + C

Question 43

Derivative of sin^-1 (u)

Answer 43

u' / sqrt(1 - u^2 )

Question 44

Derivative of tan^-1 (u)

Answer 44

u' / (1 + u^2 )

Question 45

Derivative of sec^-1 (u)

Answer 45

(u') / ( |u| )(sqrt(u^2 - 1) )

Question 46

Pythagorean Identity of cot^2(theta)

Answer 46

csc^2(theta) - 1

Question 47

Pythagorean Identity of csc^2(theta)

Answer 47

cot^2(theta) + 1

Question 48

circle identity of sin(theta)

Answer 48

y/r

Question 49

circle identity of cos(theta)

Answer 49

x/r

Question 50

circle identity of tan(theta)

Answer 50

y/x

Question 51

circle identity of csc(theta)

Answer 51

r/y

Question 52

circle identity of sec(theta)

Answer 52

r/x

Question 53

circle identity of cot(theta)

Answer 53

x/y

Question 54

cofunction identity (ie pi/2 - theta) of sin(theta)

Answer 54

cos(pi/2 - theta)

Question 55

cofunction identity (ie pi/2 - theta) of cos(theta)

Answer 55

sin(pi/2-theta)

Question 56

cofunction identity (ie pi/2 - theta) of tan(theta)

Answer 56

cot(pi/2-theta)

Question 57

cofunction identity (ie pi/2 - theta) of cot(theta)

Answer 57

tan(pi/2-theta)

Question 58

cofunction identity (ie pi/2 - theta) of sec(theta)

Answer 58

csc(pi/2-theta)

Question 59

cofunction identity (ie pi/2 - theta) of csc(theta)

Answer 59

sec(pi/2-theta)

Question 60

what does sin(theta) equal in the unit circle

Answer 60

y

Question 61

what does cos(theta) equal in the unit circle

Answer 61

x

Question 62

what does tan(theta) equal in the unit circle

Answer 62

y/x

Question 63

even odd property of sin(-theta)

Answer 63

-sin(theta)

Question 64

even odd property of -sin(theta)

Answer 64

sin(-theta)

Question 65

even odd property of cos(-theta)

Answer 65

cos(theta)

Question 66

even odd property of cos(theta)

Answer 66

cos(-theta)

Question 67

even odd property of -cos(theta)

Answer 67

-cos(theta) ie unchanged

Question 68

even odd property of tan(-theta)

Answer 68

-tan(theta)

Question 69

even odd property of -tan(theta)

Answer 69

tan(-theta)

Question 70

even odd property of csc(-theta)

Answer 70

-csc(theta)

Question 71

even odd property of sec(-theta)

Answer 71

sec(theta)

Question 72

even odd property of sec(theta)

Answer 72

sec(-theta)

Question 73

even odd property of cot(-theta)

Answer 73

-cot(theta)

Question 74

even odd property of -cot(theta)

Answer 74

cot(-theta)

Question 75

how do you find sin/cos/tan etc of (pi/x)?

Answer 75

draw a triangle with the angles (45x45x90, 30x60x90), label with unit circle values, solve pyth theorem, use soh cah toa.

Question 76

sin'(x) =

Answer 76

x'cos(x) + C

Question 77

cos'(x) =

Answer 77

-x'sin(x) + C

Question 78

tan'(x) =

Answer 78

x'sec^2(x) + C

Question 79

sec'(x) =

Answer 79

x'sec(x)tan(x) + C

Question 80

csc'(x) =

Answer 80

-x'csc(x)cot(x) + C

Question 81

cot'(x) =

Answer 81

-x'csc^2(x) + C

Question 82

if f(x) is position what is f'(x)

Answer 82

velocity

Question 83

if f(x) is position what is f''(x)

Answer 83

acceleration

Question 84

if f(x) is position what is | f'(x) |

Answer 84

speed

Question 85

arcsin'(x) =

Answer 85

1/sqrt(1-x^2) + C

Question 86

arccos'(x) =

Answer 86

-1/sqrt(1-x^2) + C

Question 87

arctan'(x) =

Answer 87

1/(1+x^2) + C

Question 88

arccsc'(x) =

Answer 88

-1/(( |x| )(sqrt(x^2 - 1)) + C

Question 89

arcsec'(x) =

Answer 89

1/(( |x| )(sqrt(x^2 - 1)) + C

Question 90

arccot'(x) =

Answer 90

-1/(1 + x^2)