chapter 6- trigonometric functions

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a function is periodic if......: its values repeat at regular intervals
if f(x) is a periodic function with period of 2, then the graph of f(x+2) is.........: identical to f(x)
If a graph of f(x) is shifted horizontally by c units and the new graph is identical to the original graph, what is the period of f(x)?: The period is c given that c is the smallest value for which f(x+c)=f(x)
What is the midline of the function and what is a formula to calculate the midline of a function?: The midline of a function is the horizontal line midway between the functions maximum and minimum values. Written as y=k where k is the midline.

A formula is:

midline = (max + min)/2
Define the amplitude and give a formula to calculate the amplitude of a function.: the amplitude is the vertical distance between the functions's maximum value (or minimum value) and the midline.

A formula for the amplitude is:

amplitude= |maximum - midline|

or |minimum - midline|
If a graph starts at x = zero and begins increasing, then is the graph more easily modeled with a sin or cos formula?: since sin(x) begins increasing immediately after x=0, then the graph is best modeled by a sin function.
What are the standard formulas for a cos graph and a sin graph?: y = Acos(B(x-h)) + k

y = Asin(B(x-h)) + k

where A = amplitude
B = 2pi/period
h = horizontal shift with respect to the original function of sin(x) or cos(x).
midline = the equation y = k
period = 2pi/B
What is B in the formula y=Asin(B(x-h))+k?: B is the angular frequency; that is the number of cycles completed from 0<=x<=2pi
True or false: The midline is a distance while the amplitude is an equation in the form y=k: False, the midline is not a distance, but rather a line y=k where k is the midpoint between the minimum and maximum values of the graph. The amplitude is a distance from the midline to the max or the min
If y = cos(3t-4) then y is the graph of y = cos(t) shifted horizontally by how many units?: Need to put equation in standard form:

y = cos(3t-4) = cos(3(t-4/3))

therefore h=4/3 so y is the graph of cos(t) shifted right by 4/3 units.
If a periodic function starts out at t=0 and begins decreasing immediately after t=0, is the function better modeled by a sin or cos function?: a cos function since y=cos(t) begins decreasing immediately after t = 0
If (x,y) is a point on the unit circle then what do x and y equal?: x=cos(theta)
y=sin(theta)

where theta is the angle that corresponds to the point (x,y) on the unit circle.
If (x,y) is a point on a circle with radius r, then what do x and y equal?: x=rcos(theta)
y=rsin(theta)

where theta is the angle that corresponds to the point (x,y)
If you are moving around the unit circle clockwise, will the angles you are sweeping out be negative or positive?: Negative
If you are moving around the unit circle counterclockwise will the angles you are sweeping out be negative or positive?: positive
If you are given x radians and asked to convert the x radians into degrees how would you do this?: multiply the x radians by 180/pi
If you are given x degrees and asked to convert these x degrees into radians how would you do this?: multiply the x degrees by pi/180
What is the arc length of a circle?: It is the measure of the arc spanned by a given angle in the unit circle.
What is the formula for the arc length?: arc length =

(radius of the circle)(angle in radians)

s = r(theta)

where s=arc length
r=radius of the circle
theta=the corresponding angle
in radians
What is the range of cos(x)? What is the domain of cos(x)?: Range: -1<=cos(x)<=1
Domain: all real numbers
What is the range of sin(x)? What is the domain of sin(x)?: Range: -1<=sin(x)<=1
Domain: all real numbers
What are the periods of both sin(x) and cos(x)?: 2pi radians or 360 degrees
How many solutions of cos(t) = 0.4 does the inverse cos function give you?: Only one solution
How do you find all the solutions of cos(t) = 0.4 for a given domain?: Use the starting value you get from the cos inverse function, then find the other solution by symmetry. After that add and subtract the period times k where k=0,1,2,....

each k that you apply will give a different solution
since cos(input)=output then inversecos(_______) = __________: output
input
find one solution to the equation using algebra: 2sin(x)=0.1: 2sin(x)=0.1
sin(x)=0.1/2
inversesin(0.1/2) = x

thus x = 0.05
When trying to solve the equation: sin(x)=0.8 you need to graph which two lines?: Graph y = sin(x)
y = 0.8

Find where these graphs intersect and these will be the solutions to the equation.
Where are the inversecos and inversesin buttons located on your calculator?: they are the functions above the cos and sin buttons and can be attained by pressing 2nd cos or 2nd sin respectively.
What are other names for inversecos and inversesin?: arccos and arcsin
Is cos(x) even, odd, or neither? What about sin(x)?: cos(x) is even
sin(x) is odd
Is a parabola a periodic function?: No, it does not repeat. It is a quadratic function.
The angle of 2pi/3 radians is in which quadrant?: 2pi/3 radians
= (2pi/3)(180/pi) = 120 degrees

Thus, 2pi/3 radians is in the 2nd quadrant
How is the period of cos(3x) related to the period of cos(x)?: since B=3 for cos(3x) then
period = 2pi/B = 2pi/3

and the period of cos(x) is 2pi.

Thus the period of cos(3x) is 1/3 that of the the period of cos(x)
Let sin(x) be transformed into sin(0.5x) +2 List the transformations in the order in which they occurred: 1. period is increased by a factor of 1/0.5 = 2. Thus the period is doubled so the graph is horizontally stretched by a factor of 2 away from the y axis.

2. the graph is then shifted vertically up by two units
The function -3cos(-x) is the graph of cos(x) when it is...: first flipped over the y axis
then vertically stretched by a factor of 3
then flipped over the x axis
What are the amplitude, period, midline of the function y=-5sin(4t-9) + 7: amplitude=|-5| = 5
period = 2pi/B and B is found by
(4t-9) = 4(t-9/4) so B = 4.
Thus the period is pi/2
midline = 7
what does arccos(cos(x)) equal?: x
what does cos(arccos(x)) equal?: x
what does sin(arcsin(x)) equal?: x
what does arcsin(sin(100x+2)) equal?: 100x + 2

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