Chapter 7 Trigonometry and Periodic Functions Flashcards
Is the average rate of change in a periodic function constant?
No. It varies according to current period
Periodic Functions
Functions with repeating values with regular intervals or periods
Period
The distance between consecutive peaks in the graph of a periodic function
Midline
The horizontal line midway between the max and min values
Amplitude
The vertical distance from midline to peak
Where on the unit circle would a zero degree angle be on?
The 3 O’clock position
From the zero angle, are degrees increments going counter clockwise or clockwise on the unit circle?
Counter-clockwise
Can positive and negative angles be larger than 360 degrees?
Yes and if so they would continue to wrap around the circle
What is the unit circle?
The circle of radius one centered at the origin point on a Cartesian graph.
How are the points of the unit circle defined?
Using the sin and cosin functions
What does the “sin” function output?
Given an angle that determines a point on the unit circle, the “sin” function determines the y-coordinate of the point on the unit circle.
What does the “cosin” function output?
Given an angle that determines a point on the unit circle, the “cosin” function determines the x-coordinate of the point on the unit circle
What is a Reference Angle?
It is the angle measured to the nearest part of the x-axis and is always between 0 and 90 degrees and is opposite of the current angle
What is a Radian?
a unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius
1rad × 180/π = 57.296°
What is the circumference of a circle?
C = 2(pi)r
Using the circumference formula, how would you calculate the value of 1 radian?
1 radian = 360 degrees/(2(pi))
Which is equal to 57.296 degrees
How would you find the radians with a given angle?
2(pi)/(given degree)
What is an arc length?
The arc length, s, spanned in a circle of radius r by an angle of (degree) in radians is
s = r(degree)
What is the domain and range of sine and cosin?
The domain is all real numbers
The range is between -1 and 1
Is Sine an odd or even function? Explain what that means
It is an odd function which means the graph is symmetrical about the origin
Is Cosine an even or odd function?
It is an even function which means that it is symmetrical about the y-axis
What is the midline, amplitude and period of the sine and cosin functions?
The midline is found on y=0
The amplitude is 1
The period is 2(pi) which is equivalent to a full 360 degrees
What is a sinusoidal function?
It is a periodic function graph showing regular oscillations
What makes a function periodic?
A function is periodic if it’s values repeat at regular intervals
What is the period of
sine(t) and sine(2t)
The period of sine(t) is 2(pi)
The period of sine(2t) is (pi)
What is the formula to find out the period?
P=2(pi)/|B|
Where B is compression or stretching value like
y=sine(Bt) or y=cosine(Bt)
What is the frequency?
The number of cycles in one unit of time
Define the various variables in the following sinusoidal function
y = A(sine(Bt) + k
“A” is the amplitude
“B” is the compression/stretching value
“k” is the midline
“t” is the period
What are the steps to finding a formula for an oscillating function?
Determine whether the graph is an odd or even function. That will determine whether it is a sine or cosine
Determine the k or midline value
Determine the A or amplitude value
Determine the Period between two max points
Use P=2(pi)/B to determine the B value
Plug all your values together
How are sine and cosin related?
They are horizontal shifts of each other
What is a phase shift?
It is a fraction of a period, more valuable in the real world, that the curve has shifted
How do you get the phase shift?
From an unfactored horizontal shift value.
g(t) = cos(3t-(pi)/4)
Factored gives the horizontal shift value
g(t) = cos(3(t - (pi)/12))
What does a positive and negative phase shift represent?
A positive phase shift value represents a shift to the left
A negative one represents a shift to the right
What is a formula to calculate the phase shift?
(2(pi) * period)
What does the tangent function represent?
The slope of the line through the origin and point (x,y)
What does the tangent function equal?
tan() = sine/cosin
What is the period of tan()?
It is pi
Does tan() have an amplitude?
No because there is no max, min, or midline values
At what points is tan() undefined and why?
Because the tan() = sine()/cosin()
Whenever cosin() = 0, tan() is undefined.
What is the Pythagorean identity?
x^2 + y^2 = 1
Cosin^2 + Sine^2() = 1
What is the secant() Function?
The inverse of cosin()
secant() = 1/cosin()
What is cosecant()?
The inverse of sine()
csc() = 1/sine()
What is the cotangent()?
The inverse of tan()
cot() = 1/tan()
In terms of pi, what is the relationship of sine and cos?
sine t = cosin(t-(pi/2)) = -sine(-t)
cosin t = sine(t+(pi/2)) = cos(-t)
What is the inverse cosine function?
Cos ^-1
Finds one of many possible angles whose cosine is a given value
Degree of angle = cos^-1(-0.4) = 1.982
What is another way to refer to the inverse of cosine?
arccos()
What is the domain of arccos?
Between 0 and pi
What is the inverse sine function?
Determine the angle with the given sine value
sin^-1
What is another way to refer to sine^-1?
Arcsine()
What is the domain of arcsine()?
Between -(pi)/2 and pi/2
What is the inverse tangent function?
tan^-1()
Gives the degree value for a given tangent value.
What is another way to refer to the tan^-1?
Arctan()
What is the domain and range of arctan()?
The domain is all real numbers
The range is between -(pi)/2 and pi/2
