4.2 / 4.3 pre calc quiz
input of t angle, cos t outputs....
'x' length of intersection
input of t angle, sin t outputs....
'y' height of intersection
120 degrees, 2π/3
(-1/2, √3/2)
135 degrees, 3π/4
(-√2/2, √2/2
150 degrees, 5π/6
(-√3/2, 1/2)
90 degrees, π/2
(0, 1)
60 degrees, π/3
(1/2, √3/2)
45 degrees, or π/4
(√2/2, √2/2)
30 degrees, or π/6 coordinate
(√3/2, 1/2)
sec t =
1/x; x can't be 0
cdc t =
1/y; y can't be 0
Cofunctions
A trigonometric function whose value for the complement of an angle is equal to the value of a given trigonometric function of the angle itself <the sine is the cofunction of the cosine> For ex, the sin of one angle is equal to the opp/hyp, while the cos of the other angle is equal to the adj/hyp, or the same exact ratio so the cos of theta= sin(90-theta)
If cotθ=1/4 and tan(π/2-θ)
Cotθ=tan(90-θ) 1/4=π/2 - θ 1/4 = definition
cosine domain and range:
D: (-inf, inf) all real numbers R: [-1, 1]
sine domain and range
D: (-inf, inf) all real numbers R: [-1, 1]
Cofunction identities
Sinθ=cos(90-θ) Tanθ=cot(90-θ) Secθ=csc(90-θ) Cosθ=sin(90-θ) Cotθ=tan(90-θ) Cscθ=sec(90-θ)
odd functions
algebra: when you input -x into a function, you get the opposite function out graphically: end behavior is going opposite way, point symmetry
even functions
algebra: when you input -x into a function, you get the same function out graphically: end behavior is going same way, symm in y-axis
pythagorean identities (3) ***
all stemmed from x^2 + y^2 = 1 -plug in sin and cos for x and y in OG and use reciprocal identities to replace OG: sin^2 t + cos^2 t = 1 1 + tan^2 t = sec^2 t 1 + cot^2 t = csc^2 t going to be pos/neg because of square root so look where it is (I II II or IV quad) to eliminate pos/neg value solving for
quotient identities (2)
combines all into 2 equations tan t = sin t / cos t cot t = cos t / sin t
trigonometric identities
equations that are always true for all real numbers for which the trigonometric function is defined
sec (-t) = sec(t)
even
cos (-t) = cos(t)
even function because if you put in a negative angle, you get the same as a positive angles output
reciprocal identities (6) sin cos tan csc sec cot
just plugging in trig terms for x and y sin t= 1/csc t cos t= 1/sec t tan t = 1/cot t csc t = 1 /sin t sec t = 1/cos t cot t= 1/tan t
unit circle angles for 30,60,90
legs: opp 30 deg: 1/2, opp 60 deg: √3/2 hyp: 1
unit circle angles for 45,45,90
legs: √2/2, √2/2 hyp: 1
cot (-t) = -cot
odd
csc (-t) = -csc t
odd
sin (-t) = -sin t
odd
tan (-t) = -tan t
odd
periodic properties of sin and cos
sin (t + 2pi) = sin t cos (t + 2pi) = cos t *both have period 2pi, so just be neg if not multiple of 2pi
periodic properties of tan and cot
tan (t + pi) = tan t cot (t + pi) = cot t *both have period pi, so just
cos t =
x
cot t =
x/y; y can't be 0
how to prove on unit circle ? (formula)
x^2 + y^2 = 1
sin t =
y
tan t =
y/x; x can't be 0
