unit #11.lesson #1.rotations and angle...

Name: ___________________________________ Date: _________________

COMMON CORE ALGEBRA II, UNIT #11 – THE CIRCULAR FUNCTIONS – LESSON #1 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

ROTATIONS AND ANGLE TERMINOLOGY COMMON CORE ALGEBRA II

In this unit we will be studying the three basic trigonometric functions. These functions are based on the geometry of a circle and rotations around its center. Sometimes the trigonometric functions are known as circular functions. In this introductory lesson we introduce some basic terminology and concepts concerning angles. Some of the terminology is specified below. Standard Position: An angle is said to be drawn in standard position if its vertex is at the origin and its initial ray points along the positive x-axis. Positive and Negative Rotations: A rotation is said to be positive if the initial ray is rotated counter-clockwise to the terminal ray and said to be negative if the initial ray is rotated clockwise to the terminal ray. Coterminal Anlges: Any two angles drawn in standard position that share a terminal ray. Reference Angles: The positive acute angle formed by the terminal ray and the x-axis. Exercise #1: For each of the following angles, given by the Greek letter theta, draw a rotation diagram and identify the quadrant is terminal ray falls in.

(a) 145 (b) 320 (c) 72 (d) 210

(e) 250 (f) 310 (g) 460 (h) 400

Reference Angle

I

IV III

II

Initial Ray

Terminal Ray

x

y


Exercise #2: In which quadrant would the terminal ray of an angle drawn in standard position fall if the angle

measures 860 ?

(1) I (3) III

(2) II (4) IV Exercise #3: Give a negative angle that is coterminal with each of the following positive angles, alpha.

(a) 90 (b) 330 (c) 120 (d) 210 Exercise #4: Coterminal angles drawn in standard position will always have measures that differ by an integer multiple of

(1) 90 (3) 180

(2) 360 (4) 720 Exercise #5: For each of the following angles, beta, draw a rotation diagram and then state beta’s reference

angle, r .

(a) 160 (b) 300 (c) 210 (d) 78

(e) 110 (f) 280 (g) 605 (h) 410

Name: ___________________________________ Date: _________________


ROTATIONS AND ANGLE TERMINOLOGY COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. For each of the following angles, draw a rotation diagram and then state the quadrant the terminal ray of the angles falls within.

(a) 135 (b) 300 (c) 110

(d) 310 (e) 85 (f) 560 2. For each of the following angles, draw a rotation diagram and determine the reference angle.

(a) 245 (b) 290 (c) 130

(d) 242 (e) 475 (f) 432


3. Give two angles that are coterminal with each of the following angles. Make one of the coterminal angles positive and one negative.

(a) 105 (b) 220 (c) 80 (d) 245

4. When drawn in standard position, which of the following angles is coterminal to one that measures 130 ?

(1) 430 (3) 850

(2) 70 (4) 730 5. Which of the following angles, when drawn in standard position, would not be coterminal with an angle that

measures 270 ?

(1) 90 (3) 630

(2) 990 (4) 720

6. Which of the following angles would not have a reference angle equal to 30 ?

(1) 210 (3) 120

(2) 330 (4) 30

REASONING 7. Angles are a measurement of rotation about a point. Are two coterminal angles the same rotation? Explain

your answer. Diagrams are helpful.

Name: ___________________________________ Date: _________________


RADIAN ANGLE MEASUREMENT COMMON CORE ALGEBRA II

Just as distance can be measured in inches, feet, miles, centimeters, etcetera, rotations about a point can also be

measured in many different ways. Measuring one complete rotation in terms of 360 is somewhat arbitrary. A common unit of angle measurement that is an alternative to degrees is called the radian. You first saw this unit of angle in Common Core Geometry. It is natural to define it in terms of the geometry of a circle, given the unit we are in: Exercise #1: Consider a full rotation around any point in the counter-clockwise (positive) direction. Radians essentially measure the total number of radii (hence the name) that have been traversed about the circumference of a circle in a given rotation. Based on the circumference formula of a circle, thus, there will always be 2 radians in one full rotation. Exercise #2: Use the formula above to answer each of the following.

THE DEFINITION OF A RADIAN

The radian angle, , created by a rotation about a point A using a radius of r and passing through an arc length of s is defined as

or equivalently s s rr

r

r

s

A

(a) What is the arc length, s, in terms of the radius of the circle, r, for a full rotation?

(b) Based on the definition above and on your answer to part (a), how many radians are there in one full rotation?

(a) Determine the number of radians that the minute hand of a clock passes through if it has a length of 5 inches and its tip travels a total distance of 13 inches.

(b) If a pendulum swings through an angle of 0.55 radians, what distance does its tip travel if it has a length of 8 feet?


Radians are essential in the study of higher-level mathematics and physics and are the angle measurement of choice for the study of calculus. It is important to be able to convert between the angular system of degrees and that of radians. The next exercise will illustrate this process.

Exercise #3: Consider one-half of a full rotation. Exercise #4: Convert each of the following common angles in degrees into radians. Express your answers in terms of pi.

(a) 90 (b) 120 (c) 225 Exercise #5: Convert each of the following common radian angles into degrees.

(a) 5

6

(b) 3

2

(c) 3

4

We should also feel comfortable with the fact that radians do not always have to be in terms of pi, although they often are.

Exercise #6: Convert each of the following radian angles, which aren’t in terms of pi, into degrees. Round your answers to the nearest degree.

(a) 5.8 (b) 4.2 (c) 2.5 Exercise #7: An angle drawn in standard position whose radian measure is 2 radians would terminate in which of the following quadrants?

(1) I (3) III (2) II (4) IV

(a) What is the angle of rotation in both degrees and in radians?

(b) Using the two equivalent angles of rotation in

(a), convert a 30 angle into an equivalent angle in radians.

Name: ___________________________________ Date: _________________


RADIAN ANGLE MEASUREMENT COMMON CORE ALGEBRA II HOMEWORK

FLUENCY 1. Convert each of the following common degree angles to angles in radians. Express your answers in exact

terms of pi.

(a) 30 (b) 45 (c) 60 (d) 180

(e) 300 (f) 135 (g) 270 (h) 330 2. Convert each of the following angles given in radians into an equivalent measure in degrees. Your answers

will be integers.

(a) 2

3

(b)

2

(c)

11

4

(d)

4

3

3. If an angle is drawn in standard position with each of the following radians angles, determine the quadrant

its terminal ray lies in. Hint – convert each angle into degrees. (a) 4.75 (b) 5.28 (c) 1.65 (d) 7.38


4. Draw a rotation diagram for each of the following radian angles, which are expressed in terms of pi. Then, determine the reference angle for each, also in terms of pi. Think back to how you did this with degrees.

(a) 2

3

(b)

11

6

(c)

5

4

APPLICATIONS 5. A dog is attached to a 10 foot leash. He travels around an arc that has a length of 25 feet. Which of the

following represents the radian angle he has rotated through?

(1) 5 (3) 2.5

(2) 7.5 (4) 1.25 6. A wheel whose diameter is 3 feet rolls a distance of 45 feet without slipping. Through what radian angle did

the wheel rotate?

(1) 30 (3) 30

(2) 25 (4) 12 7. The distance from the center of a Ferris wheel to a person who is riding is 38 feet. What distance does a

person travel if the Ferris wheel rotates through an angle of 4.25 radians?

(1) 80.75 feet (3) 507 feet

(2) 42.5 feet (4) 161.5 feet 8. A golfer swings a club about a pivot point. If the head of the club travels a distance of 26 feet and rotates

through an angle of 5 radians, which of the following gives the distance the club head is from the pivot point?

(1) 1.7 feet (3) 5.2 feet

(2) 2.6 feet (4) 7.2 feet

Name: ___________________________________ Date: _________________


THE UNIT CIRCLE COMMON CORE ALGEBRA II

The basis of trigonometry will be a very special circle known as the unit circle. This is simply a circle that has its center located at the origin and has a radius equal to one unit (hence the name "unit"). Exercise #1: From our work with equations of circles, which of the following would represent the equation of the unit circle?

(1) 1x y (3) 2 2 1x y

(2) 2 1y x (4) 2 21 1 1x y

Next we will seek to produce some of the coordinate points that lie on the unit circle through the use of the Pythagorean Theorem. The next two exercises will illustrate the important right triangles we will need.

Exercise #2: Consider the right triangle shown whose hypotenuse is equal

to one and whose angles are both equal to 45 . Since this is an isosceles right triangle, the two equal sides are labeled x. Solve for x and place your answer in simplest radical form.

Exercise #3: Consider the 30 - 60 right triangle shown whose hypotenuse is equal to one. Clearly this triangle is half of an equilateral triangle. (a) What is the length of the shorter side of this right triangle? (b) Using the Pythagorean Theorem, find the length of the longer side in

simplest radical form.

60

30

1

x

x 1

45

45


Exercise #4: The diagram below represents the unit circle. Based on your work from Exercises #2 and #3, fill in the ordered pairs at each of the following angles that are assumed to be drawn in standard position.

Exercise #4: For each of the following angles drawn in standard position, give the coordinate pair from the unit circle.

(a) 120 (b) 495 (c) 3

(d)

3

2

30

60

0

45

150

120

135

90

180

270

240

210

225

300

330

315

Name: ___________________________________ Date: _________________


THE UNIT CIRCLE COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Draw a rotation diagram for each of the following angles and then determine the ordered pair that lies on the unit circle for each angle.

(a) 330 (b) 135 (c) 270

(d) 240 (e) 540 (f) 300 2. Draw a rotation diagram for each of the following radian angles and then determine the ordered pair that lies

on the unit circle for each angle.

(a) 2

3

(b) 3

2

(c) 11

6

(d) 2

(e) 3

4

(f) 4

3


3. All of the points on the unit circle must satisfy the equation 2 2 1x y . Verify that this equation is true for

each of the coordinate points given below.

(a) 1, 0 (b) 1 3

,2 2

(c) 2 2

,2 2

4. There are other points on the unit circle besides the ones that we determined in this lesson. Every point,

though, must satisfy the equation 2 2 1x y . For each of the following problems, either the x or y

coordinate of a point on the unit circle is given. Find all possibilities for the other coordinate for this point using the unit circle equation.

(a) 3

5x (b)

5

13y (c)

1

4x

5. For each of the following angles, determine its reference angle. Then state the coordinate on the unit circle

for both the angle and its reference. What do you notice about the coordinate pairs?

(a) 150 (b) 225 (c) 300

COMMON CORE ALGEBRA II, UNIT #11 – THE CIRCULAR FUNCTIONS eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

THE UNIT CIRCLE

3 1,

2 2

30 ,6

2 2,

2 2

454

,

1 3,

2 2

60 ,3

0,1

902

,

1 3,

2 2

2120

3,

2 2,

2 2

3135 ,

4

3 1,

2 2

5

150 ,6

1, 0

180 , 0 , 360 , 2

1, 0

3 1,

2 2

7210 ,

6

2 2,

2 2

5225 ,

4

1 3,

2 2

4240 ,

3

0, 1

3270

2,

1 3,

2 2

5300 ,

3

2 2

,2 2

7315 ,

4

3 1,

2 2

11330 ,

6

Name: ___________________________________ Date: _________________


THE DEFINITION OF THE SINE AND COSINE FUNCTIONS COMMON CORE ALGEBRA II

The sine and cosine functions form the basis of trigonometry. We would like to define them so that their definition is consistent with what you already are familiar with concerning right triangle trigonometry. Recall from Common Core Geometry that in a right triangle the sine and cosine ratios were defined as:

side length opposite of sin

length of the hypotenuse

AA and

side length adjacent to cos

length of the hypotenuse

AA

Exercise #1: Consider the unit circle shown below with an angle, , drawn in standard position.

(a) Given the right triangle shown, find an expression for sin .

(b) Given the right triangle shown, find an expression for cos .

We thus define the sine and cosine functions by using the coordinates on the unit circle. They are the first functions that are geometrically defined as they are based on the geometry of a circle (circular functions).

The above definition is unquestionably the most important fact to memorize concerning trigonometry. We can now use this along with our work on the unit circle to determine certain exact values of cosine and sine.

Exercise #2: Using the unit circle diagram, determine each of the following values.

(a) sin 30 (b) sin 240 (c) cos 90 (d) cos 180

(e) sin 90 (f) sin 135 (g) cos 150 (h) cos 0

,x y

y

x

1 y

x

THE DEFINITION OF THE SINE AND COSINE FUNCTIONS

For an angle in standard position whose terminal ray passes through the point ,x y on the unit circle:

sin the -coordinatey and cos the -coordinatex


Exercise #3: The terminal ray of an angle, , drawn in standard position passes through the point 0.6, 0.8 .

Which of the following gives the value of sin ?

(1) 1.2 (3) 0.6

(2) 0.8 (4) 0.2 It is important to be able to determine the sign (positive or negative) of each of the two basic trigonometric functions for an angle whose terminal ray lies in a given quadrant. The next exercise illustrates this process.

Exercise #4: For each quadrant below, determine if the sine and cosine of an angle whose terminal ray falls in

the quadrant is positive or negative .

Since each point on the unit circle must satisfy the equation 2 2 1x y , we can now state what is known as the

Pythagorean Identity.

Exercise #5: An angle, , has a terminal ray that falls in the second quadrant. It is known that 3sin

5 ,

determine the value of cos .

Exercise #6: An angle, , has a terminal ray that falls in the first quadrant and 1cos

3 . Determine the

value of sin in simplest radical form.

I II III IV

sin

cos

THE PYTHAGOREAN IDENTITY

For any angle, , 2 2cos sin 1

Name: ___________________________________ Date: _________________


THE DEFINITION OF THE SINE AND COSINE FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Which of the following is the value of sin 60 ?

(1) 2

2 (3)

3

2

(2) 1

2 (4)

2

3

2. Written in exact form, cos 135 ?

(1) 1

2 (3)

3

2

(2) 2

2 (4)

4

3. Which of the following is not equal to sin 270 ?

(1) cos 180 (3) sin 90

(2) cos 0 (4) sin 360

4. The terminal ray of an angle drawn in standard position passes through the point 0.28, 0.96 , which lies

on the unit circle. Which of the following represents the sine of this angle?

(1) 0.96 (3) 0.28

(2) 0.68 (4) 0.29

5. The point 5,12A lies on the circle whose equation is 2 2 169x y . Which of the following would

represent the cosine of an angle drawn in standard position whose terminal rays passes through A?

(1) 5 (3) 5

13

(2) 5

12 (4) 12


6. Which of the following values cannot be the sine of an angle? Hint, think about the range of y-values on the unit circle.

(1) 7

13 (3)

3

2

(2) 5

3 (4)

11

4

7. For an angle drawn in standard position, it is known that its cosine is negative and its sine is positive. The terminal ray of this angle must terminate in which quadrant?

(1) I (3) III

(2) II (4) IV

8. If both the sine and cosine of an angle are less than zero, then when drawn in standard position in which quadrant would the terminal ray fall?

(1) I (3) III

(2) II (4) IV

9. Which of the following has a cosine that is different from sin 30 ?

(1) 60 (3) 60

(2) 300 (4) 120 10. When drawn in standard position, an angle has a terminal ray that lies in the second quadrant and whose

sine is equal to 9

41. Find the cosine of in rational form (as a fraction).

11. If the terminal ray of lies in the fourth quadrant and 3sin

3 determine cos in simplest form.

Name: ___________________________________ Date: _________________


MORE WORK WITH THE SINE AND COSINE FUNCTIONS COMMON CORE ALGEBRA II

The sine and cosine functions are the first a student typically encounters that are non-algebraic, that is they cannot be thought of as combinations of a finite number of integer powers and/or roots. Since they are defined by using the geometry of a circle they are not all that intuitive. Additional practice will be given in this lesson to simply get used to them. Exercise #1: Recall the following definitions of the sine and cosine functions. If is an angle drawn in

standard position whose terminal ray passes through the point ,x y on the unit circle then …

sin and cos

Exercise #2: Given the function 6sinf x x which of the following is the value of 60f ?

(1) 4 2 (3) 3

(2) 3 3 (4) 0

Exercise #3: If 4cos 2sing then 330g ?

(1) 8 2 3 (3) 2 3 1

(2) 4 3 2 (4) 6 2 4

Exercise #4: Which of the following is not equal to one?

(1) cos 0 (3) sin 90

(2) cos 360 (4) sin 270

Exercise #5: For an angle whose terminal ray lies in the third quadrant it is known that cos 0.96 .

Which of the following is the value of sin ?

(1) 0.28 (3) 0.04

(2) 0.56 (4) 0.78


A special relationship exists between the trigonometric values of an angle and those of its reference angle. This important relationship is illustrated in the next exercise.

Exercise #6: In each of the following, an angle and its reference have been given. Using your calculator, in degree mode, determine the sine and cosine of both the angle and its reference. Round all answers to the nearest hundredth.

(a) 110 and 70r (b) 235 and 55r (c) 282 and 78r

Clearly the absolute value of the sine and cosine are the same for an angle and its reference. This fact can be exploited to produce sine and cosine values for angles if they are known for their references.

Exercise #7: Given that 3 1cos 30 and sin 302 2 , determine the following values in exact form.

(a) cos 150 (b) sin 150 (c) cos 210

(d) sin 210 (e) cos 330 (f) sin 330

We should not forget that the trigonometric functions are valid for radians as well as degrees. Practice evaluating these functions for each of the following radian inputs. If needed, convert to degrees.

Exercise #8: Evaluate each of the following trigonometric expressions.

(a) sin2

(b) sin3

(c) 3

cos2

(d) 3

cos4

Name: ___________________________________ Date: _________________


MORE WORK WITH THE SINE AND COSINE FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. If 10sin 3f x x then 30f ?

(1) 3 32 (3) 52

(2) 2 (4) 343 2

2. If 2 and cos then 2

f x x g x x g f

?

(1) 1 (3) 0

(2) 22 (4) 1

3. Which of the following represents a rational number?

(1) sin6

(3) cos4

(2) 2

sin3

(4) 5

cos4

4. When drawn in standard position, an angle has a terminal ray that lies in the third quadrant. It is known

that 8cos

17 . Which of the following represents the value of sin ?

(1) 9

17 (3)

15

17

(2) 8

9 (4)

7

9

5. Which of the following is equal to sin 300 ?

(1) sin 60 (3) sin 60

(2) sin 30 (4) sin 30


6. For an angle it is known that its reference angle has a sine value of 4

5. If the terminal ray of , when

drawn in standard position, falls in the third quadrant then what is the value of cos ?

(1) 3

5 (3)

4

5

(2) 3

4 (4)

5

3

7. The point E 7, 24 lies on the circle whose equation is 2 2 625x y . If an angle is drawn in standard

position and its terminal ray passes through E, what is the value of the sine of this angle?

(1) 7 (3) 24

(2) 7

24 (4)

24

25

8. If it is known that 3 1

sin and cos3 2 3 2

then find the value of each of the following. To begin,

first determine in which quadrant each angle’s terminal ray lies.

(a) 2

sin3

(b) 4

cos3

(c) 5

sin3

(d) 2

cos3

9. Which of the following could not be the value of sin ? Explain how you can tell.

(1) 11

13 (3)

34

5

(2) 23

5 (4)

1

2

10. A person on a Ferris wheel sits a distance of 45 feet from the Ferris wheel’s center. If they are at an angle of

120 , when measured in standard position, then how high above the center of the wheel are they, to the nearest foot?

(1) 39 feet (3) 23 feet

(2) 12 feet (4) 32 feet

Name: ___________________________________ Date: _________________


BASIC GRAPHS OF SINE AND COSINE COMMON CORE ALGEBRA II

The sine and cosine functions can be easily graphed by considering their values at the quadrantal angles, those

that are integer multiples of 90 or 2

radians. Due to considerations from physics and calculus, most

trigonometric graphing is done with the input angle in units of radians, not degrees.

Exercise #1: Consider the functions sin and cosf x x g x x , where x is an angle in radians.

(a) By using the unit circle, fill out the table below for selected quadrantal angles.

x 2 32

2 0 2

32

2

cos x

sin x

(b) Graph both the sine and cosine curves on the grid shown below. Clearly label which curve is which.

32

2 2

2

32

2

y

x

(c) The domain and range of the sine and cosine functions are the same. State them below in interval notation.

Domain: Range:

(d) After how much horizontal distance will both sine and cosine repeat its basic pattern? This is called the period of the trigonometric graph. Because these graphs have patterns that repeat they are called periodic.


Now we would like to explore the effect of changing the coefficient of the trigonometric function. In essence we would like to look at the graphs of functions of the forms:

sin and cosy A x y A x

Exercise #2: The grid below shows the graph of cosy x . Use your graphing calculator to sketch and label

each of the following equations. Be sure your calculator is in RADIAN MODE.

3cosy x

4cosy x

3cos

2y x

2cosy x

As we can see, this coefficient controls the height that the cosine curves rises and falls above the x-axis. Its absolute value is given the name amplitude. In terms of sound waves it indicates the volume of the sound. Exercise #4: The basic sine function is graphed below. Without the use of your calculator, sketch each of the following sine curves on the axes below.

2siny x

4siny x

3siny x

1sin

2y x

32

2 2

2

32

2

y

x

32

2 2

2

32

2

y

x

Name: ___________________________________ Date: _________________


BASIC GRAPHS OF SINE AND COSINE COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. On the grid below, sketch the graphs of each of the following equations based on the basic sine function.

siny x

3siny x

siny x

5siny x

7sin

2y x

2. On the grid below, sketch the graphs of each of the following equations based on the basic cosine function.

cosy x

4cosy x

3cosy x

2.5cosy x

5.5cosy x

32

2 2

2

32

2

y

x

32

2 2

2

32

2

y

x


3. Which of the following represents the range of the trigonometric function 7siny x ?

(1) 7, 7 (3) 0, 7

(2) 7, 7 (4) 7, 7

4. Which of the following is the period of cosy x ?

(1) (3) 2

(2) 2 (4) 3

2

5. Which of the following equations describes the graph shown below?

(1) 3cosy x

(2) 3cosy x

(3) 3siny x

(4) 3siny x

6. Which of the following equations represents the periodic curve shown below?

(1) 4cosy x

(2) 4cosy x

(3) 4siny x

(4) 4siny x

7. Which of the following lines when drawn would not intersect the graph of 6siny x ?

(1) 8x (3) 4y

(2) 3x (4) 9y

y

x 3 22 2 2 3 2 2

y

x 3 22 2 2 3 2 2

Name: ___________________________________ Date: _________________


VERTICAL SHIFTING OF SINUSOIDAL GRAPHS COMMON CORE ALGEBRA II

Any graph primarily comprised of either the sine or cosine function is known as sinusoidal. These graphs can be stretched vertically, as we saw in the last lesson. Other transformations can occur as well. Today we will explore graphs of equations of the form:

sin and cosy A x C y A x C

Since we already understand the effect of A on the graph, it is now time to review the effect of adding a constant to an equation.

Exercise #1: Consider the function sin 3f x x .

(a) How would the graph of siny x

be shifted to produce the graph of

f x ?

(b) On the grid to the right is the basic

sine curve, siny x . On the same

grid, sketch the graph of f x .

Exercise #2: Consider the function 2cos 1y x .

(a) Using your calculator, sketch the graph on the grid to the right.

(b) Give the equation of a horizontal line

that this curve rises and falls two units above. Sketch this line on the graph.

(c) State the range of this trigonometric

function in interval notation.

32

2 2

2

32

2

y

x

32

2 2

2

32

2

y

x


For curves that have the general form sin and cosy A x C y A x C the value C is called the midline

or average value of the trigonometric function. It is the height or horizontal line that the sinusoidal curve rises

and falls above and below by a distance of A (the amplitude).

Exercise #3: Sketch and label the functions 4sin 2 and 2cos 3y x y x on the grid below. Try

them first without your calculator and then use it to help or verify your graphs. Then, state the ranges of each of the equations in interval notation.

Range of 4sin 2y x :

Range of 2cos 3y x :

Exercise #4: Determine the range of each of the following trigonometric functions. Express your answer in interval notation.

(a) 7sin 4y x (b) 5cos 2y x (c) 25sin 35y x

Exercise #5: The graph below shows a sinusoidal curve of the form siny A x C . Determine the values of

A and C. Show how you arrived at your results.

32

2 2

2

32

2

y

x

, 222

, 62

x

y

Name: ___________________________________ Date: _________________


VERTICAL SHIFTING OF SINUSOIDAL GRAPHS COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Sketch each of the following equations on the graph grid below. Label each with its equation.

4sin 2y x

2cos 4y x

sin 4y x

2. Graph and label both of the curves below. Then, state their intersection points (in other words, solve the

system of equations shown below).

4cos 1y x

cos 4y x

Intersection Points:

32

2 2

2

32

2

y

x

32

2 2

2

32

2

y

x


4. The following graph can be described using an equation of the form cosy A x C . Determine the values

of A and C. Show how you arrived at your answers.

5. The following graph can be described using an equation of the form siny A x C . Determine the values

of A and C. Show how you arrived at your answers. 6. State the range of each of the following sinusoidal functions in interval form.

(a) 10sin 3y x (b) 8cos 2y x (c) 22sin 30y x

7. When graphed, the line 14y would not intersect the graph of which of the following functions?

(1) 5cos 9y x (3) 2sin 15y x

(2) 6cos 10y x (4) 3sin 20y x

8. Which of the following functions has a maximum value of 25?

(1) 25sin 12y x (3) 8cos 17y x

(2) 10cos 35y x (4) 5sin 15y x

0, 9 2 , 9

, 5 , 5

2 , 9

x

y

2 , 30

3 2 , 6 2 , 6

3 2 , 30

x

y

Name: ___________________________________ Date: _________________


32

22

2

32

2

y

x

THE FREQUENCY AND PERIOD OF A SINUSOIDAL GRAPH COMMON CORE ALGEBRA II

A final transformation will allow us to horizontally stretch and compress sinusoidal graphs. It is important to be able to do this, especially when modeling real-world phenomena, because most periodic functions do not have a period of 2 . The first exercise will illustrate the pattern.

Exercise #1: On the grid below is a graph of the function 3cosy x .

(a) Using your calculator, sketch the

graph of 3cos 2y x on the same

axes. (b) How many full cycles or periods of

this function now fit within 2 radians?

The period, P, of a sinusoidal function is an extremely important concept. It is defined as the minimum horizontal shift needed for the function to repeat its fundamental pattern. The period for the basic sinusoidal graphs is 2 . Clearly, from our first exercise, the period of the function depends on the coefficient B

in the general equations sin and cosy A Bx y A Bx . This coefficient, B, is known as the frequency.

Exercise #2: Consider the graphs from Exercise #1. For each below, state the frequency and period.

(a) 3cosy x (b) 3cos 2y x (c) 1

3cos2

y x

Frequency, B Frequency, B Frequency, B Period, P Period, P Period, P

(c) Using your calculator, sketch the graph of

13cos

2y x

on the same axes.

(d) How many full cycles or periods of this function now fit within 2 radians?


Clearly we can see from Exercise #2 that the frequency and period are inversely related, that is as one increases the other decreases and vice versa.

Exercise #3: Examine the results from Exercise #2. What is true about the product of the period, P, and the frequency, B? Write an equation for this relationship. Exercise #4: Determine the period of each of the following sinusoidal functions. Express your answers in exact form.

(a) 6sin 4y x (b) 8cos3

y x

(c) 2

12sin3

y x

Exercise #5: Sketch the function 2sin 4y x on the grid below for one full period to the left and right of the

y-axis. Label the scale on your axes.

Exercise #6: The heights of the tides can be described using a sinusoidal model of the form cosy A Bx C .

If high tides are separated by 24 hours, which of the following gives the frequency, B, of the curve?

(1) 12 (3) 12

(2) 24

(4)

6

y

x

Name: ___________________________________ Date: _________________


THE FREQUENCY AND PERIOD OF A SINUSOIDAL GRAPH COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. For each of the following sinusoidal functions, determine its period in exact terms of pi.

(a) 6sin 10y x (b) 2cos 8y x (c) 1

7sin3

y x

(d) 2 4

cos3 3

y x

(e) 8sin 0.25y x (f) 2.5cos 0.4y x

2. For each of the following sinusoidal functions, determine its exact period.

(a) 2

5sin7

y x

(b)

25cos 12

365y t

(c) 8sin 19

y x

3. If the period of a sinusoidal function is equal to 18, which of the following gives its frequency?

(1) 9

(3)

18

(2) 18 (4) 6

4. It is known for that a particular sine curve repeats its fundamental pattern after every 2

7

units along the x-

axis. Which of the following is the frequency of this curve?

(1) 2

7 (3)

7

2

(2) 7 (4) 14


5. When the period of a sine function doubles the frequency

(1) doubles. (3) is halved.

(2) increases by 2. (4) decreases by 2. 6. Which of the following graphs shows the relationship between the frequency, B, and the period, P, of a

sinusoidal graph? Experiment on your calculator. Graph the expression 2

PB

.

(1) (3) (2) (4)

7. Consider the curve whose equation is 2cos 38

y x

.

(a) Determine the exact period of this sinusoidal function.

(b) What is the amplitude of this

sinusoidal function? (c) What is the midline value of this

sinusoidal function? (d) Sketch the function on the axes

for a full period on both sides of the y-axis. Label the scale on your x-axis.

P

B

P

B

P

B

P

B

y

x

Name: ___________________________________ Date: _________________


SINUSOIDAL MODELING COMMON CORE ALGEBRA II

The sine and cosine functions can be used to model a variety of real-world phenomena that are periodic, that is, they repeat in predictable patterns. The key to constructing or interpreting a sinusoidal model is understanding the physical meanings of the coefficients we’ve explored in the last three lessons.

Exercise #1: The tides in a particular bay can be modeled with an equation of the form cosd A Bt C ,

where t represents the number of hours since high-tide and d represents the depth of water in the bay. The maximum depth of water is 36 feet, the minimum depth is 22 feet and high-tide is hit every 12 hours. (a) On the axes, sketch a graph of this scenario for two

full periods. Label the points on this curve that represent high and low tide.

(b) Determine the values of A, B, and C in the model.

Verify your answers and sketch are correct on your calculator.

(c) Tanker boats cannot be in the bay when the depth of water is less or equal to 25 feet. Set up an inequality

and solve it graphically to determine all points in time, t, on the interval 0 24t when tankers cannot be in the bay. Round all times to the nearest tenth of an hour.

SINUSOIDAL MODEL COEFFICIENTS

For sin and cosy A Bx C y A Bx C

A the amplitude or distance the sinusoidal model rises and falls above its midline

C the midline or average y-value of the sinusoidal model

B the frequency of the sinusoidal model – related to the period, P, by the equation 2BP

P the period of the sinusoidal model – the minimum distance along the x-axis for the cycle to repeat

d (ft)

t (hrs)


Exercise #2: The height of a yo-yo above the ground can be well modeled using the equation

1.75cos 2.25h t , where h represents the height of the yo-yo in feet above the ground and t represents

time in seconds since the yo-yo was first dropped from its maximum height. Exercise #3: A Ferris wheel is constructed such that a person gets on the wheel at its lowest point, five feet above the ground, and reaches its highest point at 130 feet above the ground. The amount of time it takes to complete one full rotation is equal to 8 minutes. A person’s vertical position, 8 minutes. A person’s vertical position, y, can be modeled as a function of time in minutes since they boarded, t, by the equation

cosy A Bt C . Sketch a graph of a person’s vertical position for one cycle and then determine the values of

A, B, and C. Show the work needed to arrive at your answers. Exercise #4: The possible hours of daylight in a given day is a function of the day of the year. In Poughkeepsie, New York, the minimum hours of daylight (occurring on the Winter solstice) is equal to 9 hours and the maximum hours of daylight (occurring on the Summer solstice) is equal to 15 hours. If the hours of daylight can be modeled using a sinusoidal equation, what is the equation’s ampltitude? (1) 6 (3) 3 (2) 12 (4) 4

(a) Determine the maximum and minimum heights that the yo-yo reaches above the ground. Show the calculations that lead to your answers.

(b) How much time does it take for the yo-yo to return to the maximum height for the first time?

y (ft)

t (min)

Name: ___________________________________ Date: _________________


SINUSOIDAL MODELING COMMON CORE ALGEBRA II HOMEWORK

APPLICATIONS 1. A ball is attached to a spring, which is stretched and then let go. The height of the ball is given by the

sinusoidal equation 4

3.5cos 55

y t

, where y is the height above the ground in feet and t is the

number of seconds since the ball was released. 2. An athlete was having her blood pressure monitored during a workout. Doctors found that her maximum

blood pressure, known as systolic, was 110 and her minimum blood pressure, known as diastolic, was 70. If

each heartbeat cycle takes 0.75 seconds, then determine a sinusoidal model, in the form siny A Bt C ,

for her blood pressure as a function of time t in seconds. Show the calculations that lead to your answer.

y (ft)

t (min)

(b) What is the maximum height the ball reaches?

(d) Draw a rough sketch of one complete period of this curve below. Label maximum and minimum points.

(a) At what height was the ball released at? Show the calculation that leads to your answer.

(c) How many seconds does it take the ball to return to its original position?


3. On a standard summer day in upstate New York, the temperature outside can be modeled using the

sinusoidal equation 11cos 7112

O t t

, where t represents the number of hours since the peak

temperature for the day.

4. The percentage of the moon’s surface that is visible to a person standing on the Earth varies with the time

since the moon was full. The moon passes through a full cycle in 28 days, from full moon to full moon. The maximum percentage of the moon’s surface that is visible is 50%. Determine an equation, in the form

cosP A Bt C for the percentage of the surface that is visible, P, as a function of the number of days, t,

since the moon was full. Show the work that leads to the values of A, B, and C. 5. Evie is on a swing thinking about trigonometry (no seriously!). She realizes that her height above the ground

is a periodic function of time that can be modeled using 3cos 52

h t

, where t represents time in

seconds. Which of the following is the range of Evie’s heights? (1) 2 8h (3) 3 5h (2) 4 8h (4) 2 5h

(a) Sketch a graph of this function on the axes below for one day.

(b) For 0 24t , graphically determine all points in time when the outside temperature is equal to 75 degrees. Round your answers to the nearest tenth of an hour.

50

90

24

O (degrees F)

t (hours)

Name: ____________________________________ Date: __________________


THE TANGENT FUNCTION COMMON CORE ALGEBRA II

The two most important circular functions are sine and cosine. But, recall from Common Core Geometry, that there was a third trigonometric function known as tangent. There are actually three more that you will learn about in future studies, but this course will only concentrate on sine, cosine, and tangent. Let’s recall how it was defined using right triangle trigonometry. Exercise #1: For each of the right triangles below, state the values of sin , cos , and tan .

(a) (b) Notice that we can have missing sides of a right triangle and still use the Pythagorean Theorem to find these ratios (many of which may involve radicals and thus irrational numbers). Just as we did with sine and cosine, we can also define tangent in terms of the unit circle. Exercise #2: Consider the unit circle shown below. Answer the following questions based on this diagram. (a) If ,x y represents a point on the circle, how can we define the

tangent function in terms of this point? (b) Given how we define sine and cosine, how can we define the tangent

function in terms of sine and cosine?

adjacent

opposite RIGHT TRIANGLE TRIGONOMETRY DEFINITION OF TANGENT

side oppositetan

side adjacent

4

5

1

2

,x y

y

x

1 y

x


Exercise #3: Use the unit circle and the definition of tangent in terms of sine and cosine to find the value for each of the following. Do not leave any complex fractions.

(a) tan 60 (b) tan 45 (c) tan 150 (d) tan 180

Because the tangent function involves division, there is a chance it could be undefined.

Exercise #4: Consider the angle 90 or 2

radians.

Exercise #5: At which of the following angles is tangent undefined?

(1) 0 (3) 120

(2) 270 (4) 180 On a final note, it is interesting that if we know sine or cosine of an angle and the quadrant of the angle we can find the other two missing trigonometric values.

Exercise #6: Determine the value of cos and tan if 5sin

13 and the terminal ray of lies in the

second quadrant. Exercise #7: If cos a , where 0a , and the terminal ray of lies in the fourth quadrant, then which of

the following gives the value of tan in terms of a.

(1) 1 a

a

(3) 21 a

(2) 21 a (4) 21 a

a

THE DEFINITION OF TANGENT IN TERMS OF SINE AND COSINE

sintan

cos

(a) State the values of sine and cosine at this angle. (b) Why would the value of tan 90 be

undefined?

Name: ____________________________________ Date: __________________


THE TANGENT FUNCTION COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Using the unit circle diagram, find the exact values for each of the following. Don’t leave any complex

fractions. Show how you arrived at your final answers. You can check using your calculator, but decimal answers should not be given. If the value of tangent is undefined, state UND.

(a) tan 60 (b) tan 150 (c) tan 225

(d) tan 270 (e) 2

tan3

(f) 11

tan6

2. The point 0.28, 0.96 lies on the unit circle. Which of the following is closest to the tangent of an angle

drawn in standard position whose terminal ray passes through this point? (1) 3.42 (3) 0.29 (2) 1.73 (4) 0.42 3. At which of the following angles is the tangent function undefined? (1) 180 (3) 45 (2) 90 (4) 360 4. Which of the following values of x is not in the domain of tan 2g x x ? Hint – you will be multiplying

each of these values by 2 before finding its tangent. (1) 45 (3) 180

(2) 0 (4) 90


5. Determine whether each function in the tables below is positive, , or negative, , for angles whose

terminal rays lie in the respective quadrants. Use values of sine and cosine to determine the sign of the tangent function.

6. For an angle it is known that tan 0 and sin 0 . The terminal ray of when drawn in standard

position must lie in which quadrant? Hint: see the table in #5 for help. (1) I (3) III (2) II (4) IV 7. For each of the following problems, the value of either sine or cosine of an angle is given along with the

quadrant in which the terminal ray of the angle lies. For each, produce the values of the two missing trigonometric function. Some of your answers will have radicals (irrational numbers) in them. You should not leave complex fractions.

I II III IV

cos

sin

tan

(a) 3cos

5A

and A terminates in quadrant II. (b) 1

sin3

and terminates in quadrant I.

(c) 5sin

13 and terminates in quadrant III. (d) 2

cos5

B and B terminates in quadrant IV.

unit #11.lesson #1.rotations and angle...

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