Chapter 7 Trigonometry and Periodic Functions

Question 1

Is the average rate of change in a periodic function constant?

Answer 1

No. It varies according to current period

Question 2

Periodic Functions

Answer 2

Functions with repeating values with regular intervals or periods

Question 3

Period

Answer 3

The distance between consecutive peaks in the graph of a periodic function

Question 4

Midline

Answer 4

The horizontal line midway between the max and min values

Question 5

Amplitude

Answer 5

The vertical distance from midline to peak

Question 6

Where on the unit circle would a zero degree angle be on?

Answer 6

The 3 O’clock position

Question 7

From the zero angle, are degrees increments going counter clockwise or clockwise on the unit circle?

Answer 7

Counter-clockwise

Question 8

Can positive and negative angles be larger than 360 degrees?

Answer 8

Yes and if so they would continue to wrap around the circle

Question 9

What is the unit circle?

Answer 9

The circle of radius one centered at the origin point on a Cartesian graph.

Question 10

How are the points of the unit circle defined?

Answer 10

Using the sin and cosin functions

Question 11

What does the “sin” function output?

Answer 11

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.

Question 12

What does the “cosin” function output?

Answer 12

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

Question 13

What is a Reference Angle?

Answer 13

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

Question 14

What is a Radian?

Answer 14

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°

Question 15

What is the circumference of a circle?

Answer 15

C = 2(pi)r

Question 16

Using the circumference formula, how would you calculate the value of 1 radian?

Answer 16

1 radian = 360 degrees/(2(pi))

Which is equal to 57.296 degrees

Question 17

How would you find the radians with a given angle?

Answer 17

2(pi)/(given degree)

Question 18

What is an arc length?

Answer 18

The arc length, s, spanned in a circle of radius r by an angle of (degree) in radians is

s = r(degree)

Question 19

What is the domain and range of sine and cosin?

Answer 19

The domain is all real numbers

The range is between -1 and 1

Question 20

Is Sine an odd or even function? Explain what that means

Answer 20

It is an odd function which means the graph is symmetrical about the origin

Question 21

Is Cosine an even or odd function?

Answer 21

It is an even function which means that it is symmetrical about the y-axis

Question 22

What is the midline, amplitude and period of the sine and cosin functions?

Answer 22

The midline is found on y=0

The amplitude is 1

The period is 2(pi) which is equivalent to a full 360 degrees

Question 23

What is a sinusoidal function?

Answer 23

It is a periodic function graph showing regular oscillations

Question 24

What makes a function periodic?

Answer 24

A function is periodic if it’s values repeat at regular intervals

Question 25

What is the period of sine(t) and sine(2t)

Answer 25

The period of sine(t) is 2(pi) The period of sine(2t) is (pi)

Question 26

What is the formula to find out the period?

Answer 26

P=2(pi)/|B| Where B is compression or stretching value like y=sine(Bt) or y=cosine(Bt)

Question 27

What is the frequency?

Answer 27

The number of cycles in one unit of time

Question 28

# Define the various variables in the following sinusoidal function y = A(sine(Bt) + k

Answer 28

“A” is the amplitude “B” is the compression/stretching value “k” is the midline “t” is the period

Question 29

What are the steps to finding a formula for an oscillating function?

Answer 29

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

Question 30

How are sine and cosin related?

Answer 30

They are horizontal shifts of each other

Question 31

What is a phase shift?

Answer 31

It is a fraction of a period, more valuable in the real world, that the curve has shifted

Question 32

How do you get the phase shift?

Answer 32

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))

Question 33

What does a positive and negative phase shift represent?

Answer 33

A positive phase shift value represents a shift to the left A negative one represents a shift to the right

Question 34

What is a formula to calculate the phase shift?

Answer 34

(2(pi) * period)

Question 35

What does the tangent function represent?

Answer 35

The slope of the line through the origin and point (x,y)

Question 36

What does the tangent function equal?

Answer 36

tan() = sine/cosin

Question 37

What is the period of tan()?

Answer 37

It is pi

Question 38

Does tan() have an amplitude?

Answer 38

No because there is no max, min, or midline values

Question 39

At what points is tan() undefined and why?

Answer 39

Because the tan() = sine()/cosin() Whenever cosin() = 0, tan() is undefined.

Question 40

What is the Pythagorean identity?

Answer 40

x^2 + y^2 = 1 Cosin^2 + Sine^2() = 1

Question 41

What is the secant() Function?

Answer 41

The inverse of cosin() secant() = 1/cosin()

Question 42

What is cosecant()?

Answer 42

The inverse of sine() csc() = 1/sine()

Question 43

What is the cotangent()?

Answer 43

The inverse of tan() cot() = 1/tan()

Question 44

In terms of pi, what is the relationship of sine and cos?

Answer 44

sine t = cosin(t-(pi/2)) = -sine(-t) cosin t = sine(t+(pi/2)) = cos(-t)

Question 45

What is the inverse cosine function?

Answer 45

Cos ^-1 Finds one of many possible angles whose cosine is a given value Degree of angle = cos^-1(-0.4) = 1.982

Question 46

What is another way to refer to the inverse of cosine?

Answer 46

arccos()

Question 47

What is the domain of arccos?

Answer 47

Between 0 and pi

Question 48

What is the inverse sine function?

Answer 48

Determine the angle with the given sine value sin^-1

Question 49

What is another way to refer to sine^-1?

Answer 49

Arcsine()

Question 50

What is the domain of arcsine()?

Answer 50

Between -(pi)/2 and pi/2

Question 51

What is the inverse tangent function?

Answer 51

tan^-1() Gives the degree value for a given tangent value.

Question 52

What is another way to refer to the tan^-1?

Answer 52

Arctan()

Question 53

What is the domain and range of arctan()?

Answer 53

The domain is all real numbers The range is between -(pi)/2 and pi/2