Angle Measure

Math 100

Section 6.1

Angle Measure

Definitions:

An angle consists of two rays, R1 and R2, with a common vertex. Often we think of an angle as being a rotation of R1 onto R2. R1 is called the initial side and R2 is called the terminal side. When the rotation is counterclockwise, the angle is said to be positive, and when the rotation is clockwise, the angle is negative.

Exercise 1 Label the initialside, terminal side and vertex of the following angles. Identify each angle as being positive or negative.

Definition:

Angle is said to be in standard position if its initial side lies along the positive x-axis an its vertex is at the origin. Label the initial side, terminal side and vertex of the angle below.

The measure of an angle is how much rotation is required to move R1 onto R2. Angles can be measured in degrees. You are probably familiar with degrees: 360 degrees is one complete revolution, 90 degrees is a right angle, and so on. Another unit used for measuring angles is radians. One radian is the angle subtended by a circular arc the length of the circle’s radius:

360 degrees = one complete revolution = 2 radians

Exercise 2 Determine how many degrees are in one radian. Sketch a angle of about one radian in standard position on the circle below.

1 radian = ______degrees

Exercise 3 Converting between radians and degrees

a) Express 90 degrees in radians

b) Express /3 radians in degrees

Exercise 4 A circle divided up into equal sized segments is shown. Fill in the blanks below, giving radian measures as multiples of  and reduce fractions to lowest terms.

Exercise 5 A circle is divided up into equal sized segments is shown. Fill in the blanks below, giving radian measures as multiples of  and reducing fractions to lowest terms.

Exercise 6 Angles in standard position are shown. Give the angle measure of each in both degrees and radians.

____ degrees = ____radians ____ degrees = ____radians

radians

____ degrees = ____radians ____ degrees = ____radians

_____ degrees = _____ radians _____ degrees = _____ radians

____ degrees = ____ radians ____ degrees = ____ radians

Coterminal Angles

Two angles in standard position are said to be coterminal if they have the same initial and terminal sides. Coterminal angles look the same after they are made, but if you watched the angles being swept out, they might appear to be different. For example, one might be positive and one negative, or one might "wrap around" more times then another.

Exercise 7 Find two angles that are coterminal with the given angle. Draw the coterminal angles in standard position.

Exercise 8 For each angle:

convert from radian to degrees—or degrees to radians

draw in standard position

state two other angles that are coterminal with the given angle

Objectives and Suggested Exercies Section 6.1

SuggestedExercises are on pages 413 - 414

Objectives

1)To convert angles between radian and degree measure (# 1 – 17 odd)

2)To draw an angle in standard position.

3)To understand the concept of coterminal angles (# 19 – 41 odd)

Math 100

Section 6.2

Trigonometry of Right Triangles

Mnemonics for remembering the trigonometric ratios:

Exercise 1 Find the exact value of the following trigonometric expressions for the given right triangle:

Sin() = ______Sin() = ______

Cos() = ______Cos() = ______

Tan() = ______Tan() = ______

Special Triangles

Two types of right triangles, 45-45-90 and 30-60-90, have trigonometric ratios that can be calculated exactly using geometry. These triangles were popular as examples in the days before calculators, and they still commonly appear in trigonometry and calculus textbooks.

Exercise 2 45 (/4 radians) Right Triangle

This triangle is obtained by drawing a diagonal in a square of side 1 (shown below).

Use the above results to find the exact value of each of the following. Verify your results by finding a decimal approximation using the trigonometric functions on your calculator:

Sin ( 45 ) = Sin ( /4 ) = ______

Cos (45 ) = Cos (/4 ) = ______

Tan (45 ) = Tan ( /4 ) = ______

Sec ( 45 ) = Sec( /4 ) = ______

Csc ( 45) = Csc( /4 ) = ______

Cot ( 45 ) = Cot ( /4 ) = ______

Exercise 3 30/60 Right Triangle (/6, /3 radians)

This triangle is obtained by drawing a perpendicular bisector in an equilateral triangle of side 2 (shown below).

Use the above results to find the exact value of each of the following. Verify your results by finding a decimal approximation using the trigonometric functions on your calculator:

Sin( 30 ) = ______Sin ( 60 ) = ______

Cos ( /6 ) = ______Cos (/3 ) = ______

Tan ( 30 ) = ______Tan ( /3 ) = ______

Exercise 4 Evaluate the following expressions exactly:

a) sin(30)cos(30) + sin(60)cos(60)

b) sin2(/3) + cos2(/3)

c) 2sin(/4)cos(/4)

Exercise 5 In each triangle, an angle and the length of one side is given. Use trigonometric functions to find the value of the unknown side x. (Not drawn to scale)

In the previous exercise the angle was given and we used the angle to find the sine, cosine or tangent.

In the following problems, we are given the sine, cosine or tangent, and we want to find the angle. The inverse trigonometric functions , pronounced as arcsin(x), arcos(x), and arctan(x), but often denoted as sin-1(x), cos-1(x), and tan-1(x), serve this purpose.

Exercise 6 Find the angle whose tangent is .5774

Exercise 7 Use inverse trig functions on your calculator to find the unknown angle  in the following triangles

Exercise 8 Sketch a triangle that has acute angle , and find the other five trigonometric ratios of . Sin  = ¾

cos = _____

tan = _____

csc = _____

sec = _____

cot = _____

Use the arctan(x) , arcsin(x), or arcos(x) function to find  in:

degrees: ______radians:______

Objectives and Suggested Exercises Section 6.2

SuggestedExercises are on pages 422 - 425

Objectives

1)To find the values of sinx, cosx, tanx, cscx, and secx when x is an acute angle in a right triangle (# 1, 3, 5)

2)To learn about the “special” angles: 45, 30 and 60 degrees (# 7, 9, 21, 23, 25)

3)To use the trig functions on your calculator to find the length of an unknown side in a triangle (#7, 9)

4)To find an unknown angle using the inverse trig functions on your calculator.

Math 100

Section 6.3

Trigonometric Functions of Angles

This section combines the ideas of section 6.1 (angle measure) and 6.2 (right-triangle trigonometry). In section 6.2, we learned how to find the trigonometric functions of angles in right-triangles (recall: SOHCAHTOA). In this section we learn how to find the trigonometric function of any angle in standard position, even those that are not in the range 0 < 90 and therefore can’t be in a right triangle.

Exercise 1 A right triangle with acute angle  and legs of length 3 and 4 is shown. Redraw the triangle so that  is in standard position--with the side adjacent to  along the positive x-axis and the right-angle away from the origin.

a) after the triangle is redrawn answer the following:

i) what is the x-coordinate of point P? x = ______

ii) what is the y-coordinate of point P? y = ______

b) what is the length of the hypotenuse? r = ______

c)find the value of the following:

i) sin  = ______ii) cos  = ______iii) tan  = ______

iv) csc  = ______v) sec  = ______vi) cot  = ______

Exercise 2 Label the hypotenuse, the side opposite , and the side adjacent to  on the right-triangle ABC. Redraw the triangle with the side adjacent to  along the positive x-axis and the right-angle away from the origin.

a) after the triangle is redrawn, label the coordinates of point B as (x,y)

Answer the rest of the questions in terms of x and y:

a)what is the length of the side opposite?______

b)what is the length of the side adjacent to ?______

b) what is the length of the hypotenuse, r?______

d)find the following:

i) sin  = ______ii) cos  = ______iii) tan  = ______

iv) csc  = _____v) sec  = ______vi) cot  = ______

The idea of the previous exercise is extended to angles that do not have a terminal arm in the first quadrant. Look at the definitions shown in the box and compare them to the results of the last exercise.

Exercise 3 Using the above definitions, find the values (where possible) of the indicated trigonometric functions for the angles shown. In each case, pick one point (call it P) on the terminal arm of the angle. Is it important which point you use?

sin  = ______P( ____ , ____ )

cos  = ______x = ____ y = ____

tan  = ______r = ______

sin  = ______P( ____ , ____ )

cos  = ______x = ____ y = ____

tan  = ______r = ______

sin  = ______P( ____ , ____ )

cos  = ______x = ____ y = ____

tan  = ______r = ______

Whether a trigonometric function is positive or negative depends only on what quadrant the terminal arm lies in.

Exercise 4 Fill in the table below, by referring to the coordinate system shown. Assume (x,y) is a point on the terminal arm of angle :

Terminal
arm in quadrant / The Sign (+ or -) of :
x / y / r / Sin  =y/r / Cos  = x/r / Tan  = y/x
I
II
III
IV

These results are summarized in the coordinate system: 

You can remember this as “all students take calculus”

Or “all schools torture children”

Or “are simpletons teaching courses?”

Reference Angles:

Exercise 5

The reference angle can be used in conjunction with knowledge about which quadrant the angle lies in to find the exact value of the trigonometric functions.

To find the exact value of trigonometric functions for any angle , carry out the following steps:

Find the reference angle associated with the angle .
Determine the sign (+ or - ) of the trigonometric function with the angle .
The value of the trigonometric function of  is the same, except possibly for sign , as the value of the trigonometric functions of .

Exercise 6

Find the exact value of the following trigonometric functions:

a)cos (5/6) reference angle = = ______cos() = ______

b)sin (3/4) reference angle = = ______sin() = ______

c)tan (-/4) reference angle = = ______tan() = ______

Exercise 7

Find the value of sin() and cos() from the information given.

Use the arctan(x) function on your calculator (tan-1x ), in conjunction with knowledge about what quadrant the terminal arm lies in, to find the approximate value of  in degrees. In each case, assume 0 <  < 2

a)tan  = 3/4, sin() > 0

sin  = ______

cos  = ______

 ______

b)tan  = 3/4, sin() < 0

sin  = ______

cos  = ______

 ______

c)tan  = -3/4, cos  < 0

sin  = ______

cos  = ______

 ______

Objectives and Suggested Exercises Section 6.3

SuggestedExercises are on pages 433 - 434

Objectives

1)To find the reference angle for a given angle. (# 1, 3, 5)

2)To find the values of trigonometric functions for angles that are not (necessarily) acute. (#7 – 29 odd)

3)To determine the quadrant in which an angle lies (#31, 33)

4)To find the values of trigonometric functions of an angle, given the value of one of the functions, and one other piece of information. (# 41 – 47 odd)

Math 100

Section 5.3

Graphs of Trigonometric Functions

Exercise 1 On an unusual new amusement park ride, riders are submerged in water (enclosed inside a water-tight car) for a portion of the ride. The ride is similar to a Ferris wheel with its bottom half underwater, as shown. The duration of the ride is 360 seconds (6 minutes ). You board the ride at water level (at the point indicated in the drawing). During the ride you are rotated counterclockwise for one complete revolution. The radius r of the ride is 1 dekametre (10 metres).

a)How many degrees do you rotate in one second?______

b)Mark your location after: 45, 90, 180, 270 and 360 seconds

c)During what time intervals will be going up?

d)During what time interval will you be going down?

e)What is the maximum elevation above the water that you will reach?

f) What is the greatest depth below the water that you will reach?

g) You reach the greatest depth ______after the ride begins.

h)During what time interval will you be above the water?

i)During what time interval will you be below the water?

In the following questions, let t be the time that has elapsed since the ride began. Recall that the radius of the ride is 1 dekametre.

a)At what times t will you reach the point labeled A, B, C and D ?

A______B______C______D______

b)Use trigonometry to estimate your elevation (in dekametres) from water-level (to 3 decimal places) at the points A, B and C. In each case, state the time t and the angle  you have rotated to arrive at the point. (When you are above the water, your elevation is positive. When you are below the water, your elevation is negative.)

i)
A t =  = ______

ii)B t =  = ______

iii) C t =  = ______

iv) D t =  = ______

Exercise 2: The sine function graph

t / Elevation = sin(t)
0
30
60
90
120
150
180
210
240
270
300
330
360

Transformations of sin(x)

Exercise 3: The graph of y = sin(x) for x from -720 to 720 (-4 to 4 radians) is given. Plot the following transformations of sin(x). In each case state the amplitute:

1. y = sin(x) + 1 amplitude:______

2. y = sin(x + 90) (in radians, y = sin(x + /2) amplitude______

y = 2sin(x) amplitude______

4. y = -2.5sin(x) amplitude______

The Graph of the Cosine Function

In the last exercise, we graphed the function f(x) = sin(x). In this exercise, we graph the cosine function. The graph of the cosine looks like the graph of the sine, only it is shifted horizontally.

Exercise 4:

Angle in radians () / Angle in degrees () / Cos()
0 / 0
/6 / 30
/3 / 60
/2 / 90
2/3 / 120
5/6 / 150
 / 180
7/6 / 210
4/3 / 240
3/2 / 270
5/3 / 300
11/6 / 330
2 / 360

The Period of a Sine Curve

The period of a sine curve is the length of time it takes to complete one complete revolution. The period of the standard sine curve f(x) = sin(x) is 360 or 2 radians. If k is a constant, the period of sin(kx) is 360/k or 2/k.

Exercise 5 Recall the amusement park ride of Exercise 1. Suppose the ride rotates 2 degrees every second (instead of 1 degree every second, as in the original question. Complete the table and answer the questions below:

a) How long will it to reach the highest point, 1 dam above the water?______

b) How long will it take to reach the lowest point?______

c) During what time interval are you going down?______

d) If the ride lasts 360 seconds, how many times will you go around?_____

Time / Degrees rotated / Elevation
0
30
60
90
120
150
180
210
240
270
300
330
360

Exercise 6 Recall again the amusement park ride of Exercise 1. Suppose the ride rotates 1/2 degrees every second (instead of 1 degree every second, as in the original question. Complete the table and answer the questions below:

a) How long will it to reach the highest point, 1 dam above the water?______

b) How long will it take to reach the lowest point?______

c) During what time interval are you going down?______

d) If the ride lasts 720 seconds, how many times will you go around?_____

Time / Degrees rotated / Elevation
0
60
120
180
240
300
360
420
480
540
600
660
720

Exercise 7 Graph each of the following functions and state the period.

a) f(x) = cos(2x) (The graph of cos(x) is given in gray.)

period = ______

b) g(x) = cos(x/2) (The graph of cos(x) is given in grey.)

period = ______

c) h(x) = sin(4x) (The graph of sin(x) is given in gray.)

period = ______amplitude______

Objectives and Suggested Exercises from Section 5.3

SuggestedExercises are on page 380

Objectives:

1)To be able to sketch the graphs of the sine and cosine functions (page 370)

2)To be able to sketch the graphs of transformations of the sine and cosine functions, in particular:

a)Vertical shifts (#1)

b)Reflections (# 2)

c)Vertical stretching (change in amplitude) (# 5)

d)Horizontal stretching (change in period) (#11, 13)

e)Horizontal shifts (#19, 21)

3)To determine the amplitude, period and horizontal shift from the graph of a sine or cosine curve, and write an equations that represents the curve (#33, 35)

Math 100

Section 7.1

Trigonometric Identities

Exercise 1 Use the trigonometric definitions to verify that the basic identities are true

a) csc() =

b) sec() =

c) cot() =

d) tan() =

e) cot() =

Exercise 2 Use the definitions to show that the first Pythagorean identity is true:

Exercise 3 Use the basic identities (not the definitions) to write each expression in terms of sines and/or cosines, and then simplify

e) (1 - sin(x))(1 + sin(x))

Exercise 4 Prove that each of the following equations is an identity:

b) (1 + sin(x))2 + cos2(x) = 2 + 2sin(x)

c) tan(x) + cot(x) = sec(x)csc(x)

Objectives and Suggested Exercises for Section 7.1

SuggestedExercises are on pages 466 - 467

Objectives:

1)To learn and understand the basic trigonometric identities. (page 461)

2)To write a trigonometric expression in terms of sine and cosine (# 1, 3, 5)

3)To simplify a trigonometric expression using the basic identities (# 7 – 19 odd)

4)To verify various trigonometric identities. (do a selection from # 21 – 81 odd)

Math 100 Chapter 6 1