Test 12
Basic Sine Graph y=sin(x)
(0, 0) (π/2, 1) (π, 0) (3π/2, -1) (2π, 0)
Basic Cosine Graph y=cos(x)
(0, 1) (π/2, 0) (π, -1) (3π/2, 0) (2π, 1)
Points in cos(x) vs cos-1(x)
(0, 1) vs (1, 0) (π/2, 0) vs (0, π/2) (π, -1) vs (-1, π)
Trigonometric Identity to know
(cosθ)^2+(sinθ)^2=1
What is the csc(3π/4)
*Just plug in 180 for π Csc(3π/4)=1/sin(135)=root(2)
Cos(A)=-3/5 and A terminates in quadrant 2. Find the other two functions.
-Draw unit circle and right triangle. -Use pythagorean theorem. -Find other functions
Frequency
How often something happens per unit of time.
How are the period and the frequency related?
the period is 2π/frequency the frequency is 2π/period
FOR y=asin(bx)
a=amplitude b=frequency
How do you find D(vertical shift)?
From the midline to the x axis!(Midline is (max+min)/2)
What is the secant of 60?
1/1/2=2
Secant
1/cosθ
Cosecant
1/sinθ
Cotangent
1/tanθ or cosθ/sinθ
What is the period of sine and cosine graphs?
2π
Which of the following values is not in the domain of g(x)=tan(2x)? Hint: You are multiplying each value by 2 before finding its tangent. 45 0 180 90
45-45*2=90
Periodic Function
A function that repeats its values in regular intervals or periods.
A=sin(B(x-C)+D
A is amplitude B is frequency(2π/p) P is period(2π/b) C is horizontal shift D is vertical shift(+ up and - down)
Sinusoidal Function
A mathematical curve that describes a smooth repetitive oscillation, named after the function sine.
What are the domains and ranges of sign and cosine functions?
Domain: (-infinity, infinity) Range: [-1, 1]
y=arctanx or y=tan-1x
Domain: All real numbers Range: (-π/2, π/2)
Tangent Graph y=tanx -Domain -Range -Points -Graph
Domain: All real numbers except odd multiples of π/2(π/2, 3π/2, 5π/2...) Range: All real numbers Amplitude: No amplitude Period: π (-2π, 0) (-3π/2, undefined) (-π, 0) (π/2, undefined) (0, 0) (π/2, undefined) (π, 0) (3π/2, undefined) (2π, 0)
y=arcsinx or y=sin-1x
Domain: [-1, 1] Range: [-π/2, π/2]
y=arccosx or y=cos-1x
Domain: [-1, 1] Range: [0, π]
How do you find C(horizontal shift)? Sine and Cosine?
For Cosine: From y axis to maximum(maximum starts on the y axis) For Sine: From y axis to midline point(midline point starts on the y axis).
State, in units of radians, the exact value for each of the following: cos-1(1/2) cos-1(-1/2) cos-1(root2/2) cos-1(-root2/2)
For the first question, for example, it is asking what angle gets cos(x)=1/2. You must find the angle using your triangles(30, 60, 90 and 45, 45, 90), then convert it to radians. cos(60)=π/3 cos(120)=2π/3 cos(45)=π/4 cos(135)=13π/4
How does amplitude change a graph? ex: y=3cos(x) and y=-1/2sin(x)
For y=3cos(x), the amplitude is simply 3 instead of 1. For y=-1/2sin(x), it goes to negative 1 first, and the amplitude is 1/2 instead of 1.
Which of the following is not in the domain of y=csc(x) 1. x=180 2. x=60 3. x=90 4. x=135
It is x=180 because the y coordinate is 0, and csc=1/sin θ
Tan60=
Sin60/Cos60=root(3)
Tangent 2 Definitions
Tanθ=side opposite/side adjacent Tanθ=y/x=sinθ/cosθ
Which of the following angles is the tangent function undefined? θ=180 θ=-90 θ=45 θ=-360
The answer is θ=-90, because that is (-1/0), and you cannot have - on the bottom
Amplitude
The height from the center line to the peak. We can measure the height from the highest to lowest points and divide that by two.
Period
The length from one peak to the next(or from any point to the next matching point).
Phase shift
The measure of horizontal shifting(always inside parentheses) -If the phase shift is a positive value of c, the horizontal shift is to the left. -If the phase shift is a negative value of c, the horizontal shift is to the right.
How do you create inverse trig graphs?
You simply reflect over the line y=x, reversing all points.
Range is always...
[-amplitude, +amplitude]
Remember, negatives...
apply to reciprocal functions as well! ASTC