Chapter 2

Copyright © 2005 Pearson Education, Inc.


Chapter 2

Acute Angles and

Right Triangles


2.1

Trigonometric Functions of Acute Angles

Copyright © 2005 Pearson Education, Inc. Slide 2-4

Right-triangle Based Definitions of Trigonometric Functions For any acute angle A in standard position.

side opposite hypotenusesin csc

hypotenuse side opposite

side adjacent hypotenusecos sec

hypotenuse side adjacent

side oppositetan

side adjacent

side adjacentcot .

side opposite

y rA A

r y

x rA A

r x

yA

x

xA

y


Example: Finding Trig Functions of Acute Angles Find the values of sin A, cos A, and tan A in the

right triangle shown.

A C

B

52

48

20

side oppositesin

hypotenuse

side adjacentcos

hypotenuse

side opposite

52

5

tanside a

20

20

4

djac

8

8ent 4

2

A

A

A


Cofunction Identities

For any acute angle A,

sin A = cos(90 A) csc A = sec(90 A)

tan A = cot(90 A) cos A = sin(90 A)

sec A = csc(90 A) cot A = tan(90 A)


Example: Write Functions in Terms of Cofunctions Write each function in

terms of its cofunction.

a) cos 38

cos 38 = sin (90 38) = sin 52

b) sec 78

sec 78 = csc (90 78) = csc 12


Example: Solving Equations

Find one solution for the equation

Assume all angles are acute angles.

cot(4 8o) tan(2 4o)

(4 8o) (2 4o) 90o

6 12o 90o

6 78o

13o

cot(4 8o) tan(2 4o) .


Example: Comparing Function Values

Tell whether the statement is true or false.

sin 31 > sin 29

In the interval from 0 to 90, as the angle increases, so does the sine of the angle, which makes sin 31 > sin 29 a true statement.


Special Triangles

30-60-90 Triangle 45-45-90 Triangle


Function Values of Special Angles

260

1145

230

csc sec cot tan cos sin

1

23

2

3

3

3 2 3

3

2

2

2

2

2 2

3

2

1

23 3

3

2 3

3


2.2

Trigonometric Functions of Non-Acute Angles


Answers to homework

Sin cos tan

1.21/29 20/29 21/20

2.45/53 28/53 45/28

3.n/p m/p n/m

4.k/z y/z k/y

5.C

6.H 9. E

7.B 10. A

8.G


Homework answers


Reference Angles

A reference angle for an angle is the positive acute angle made by the terminal side of angle and the x-axis.


Example: Find the reference angle for each angle.

a) 218 Positive acute angle

made by the terminal side of the angle and the x-axis is 218 180 = 38.

1387 Divide 1387 by 360 to get

a quotient of about 3.9. Begin by subtracting 360 three times. 1387 – 3(360) = 307.

The reference angle for 307 is 360 – 307 = 53


Example: Finding Trigonometric Function Values of a Quadrant Angle

Find the values of the trigonometric functions for 210.

Reference angle:

210 – 180 = 30Choose point P on the terminal side of the angle so the distance from the origin to P is 2.


Example: Finding Trigonometric Function Values of a Quadrant Angle continued

The coordinates of P are x = y = 1 r = 2

3, 1

3

1 3 3sin 210 cos210 tan 210

2 2 3

2 3csc210 2 sec210 cot 210 3

3


Finding Trigonometric Function Values for Any Nonquadrantal Angle Step 1If > 360, or if < 0, then find a

coterminal angle by adding or subtracting 360 as many times as needed to get an

angle greater than 0 but less than 360. Step 2Find the reference angle '. Step 3Find the trigonometric function values for

reference angle '. Step 4Determine the correct signs for the values

found in Step 3. (Use the table of signs in section 5.2, if necessary.) This gives the

values of the trigonometric functions for angle .


Example: Finding Trig Function Values Using Reference Angles Find the exact value of

each expression. cos (240) Since an angle of 240

is coterminal with and angle of 240 + 360 = 120, the reference angles is 180 120 = 60, as shown.

cos( 240 ) cos120

1cos60

2


HomeworkPg 59-60 # 1-6, 10-13


Example: Evaluating an Expression with Function Values of Special Angles Evaluate cos 120 + 2 sin2 60 tan2 30.

Since

cos 120 + 2 sin2 60 tan2 30 =

1 3 3cos 120 , sin 60 , and tan 30 ,

2 2 3

2 2

+ 2

1 3 32

2

1

4

3

3

2

9

3

2 3

2




Example: Using Coterminal Angles

Evaluate each function by first expressing the function in terms of an angle between 0 and 360.

cos 780 cos 780 = cos (780 2(360) = cos 60

= 1

2


2.3

Finding Trigonometric Function Values Using a Calculator


Function Values Using a Calculator

Calculators are capable of finding trigonometric function values.

When evaluating trigonometric functions of angles given in degrees, remember that the calculator must be set in degree mode.

Remember that most calculator values of trigonometric functions are approximations.


Example: Finding Function Values with a Calculator a) Convert 38 to decimal

degrees.

b) cot 68.4832 Use the identity

cot 68.4832 .3942492

38 38

sin 38 24 si

38.4

38.4

2424

6

n

.6211477

0

1cot .

tan

sin 38 24

24


Angle Measures Using a Calculator

Graphing calculators have three inverse functions.

If x is an appropriate number, then

gives the measure of an angle whose sine, cosine, or tangent is x.

1 1sin ,cos , x x-1 or tan x


Example: Using Inverse Trigonometric Functions to Find Angles Use a calculator to find an angle in the

interval that satisfies each condition.

Using the degree mode and the inverse sine

function, we find that an angle having sine value .8535508 is 58.6 .

We write the result as

[0 ,90 ]

sin .8535508

1sin .8535508 58.6


Example: Using Inverse Trigonometric Functions to Find Angles continued

Use the identity Find the

reciprocal of 2.48679 to get

Now find using the inverse cosine function. The result is 66.289824

sec 2.486879

1cos .

sec

cos .4021104.


Page 64 # 22-29

22. 57.99717206

23. 55.84549629

24. 81.166807334

25. 16.16664145

26. 30.50274845

27. 38.49157974

28. 46.17358205

29. 68.6732406


2.4

Solving Right Triangles


Significant Digits for Angles

A significant digit is a digit obtained by actual measurement. Your answer is no more accurate then the least accurate

number in your calculation.

Tenth of a minute, or nearest thousandth of a degree

5

Minute, or nearest hundredth of a degree4

Ten minutes, or nearest tenth of a degree3

Degree2

Angle Measure to Nearest:Number of Significant Digits


Example: Solving a Right Triangle, Given an Angle and a Side Solve right triangle ABC, if A = 42 30' and c = 18.4. B = 90 42 30'

B = 47 30'

AC

B

c = 18.4

4230'

sin A a

c

sin 42o30' a

18.4

a 18.4sin 42o30'

a 18.4(.675590207)

a 12.43

cos A b

c

cos42o30' b

18.4

b 18.4cos42o30'

b 13.57


Example: Solving a Right Triangle Given Two Sides Solve right triangle ABC if a = 11.47 cm and c = 27.82 cm.

B = 90 24.35B = 65.65

AC

B

c = 27.82a = 11.47

1

side oppositesin

hypotenuse

11.47.412293314

27.82

sin 24.35

A

A

2 2 2

2 2 227.82 11.47

25.35

b c a

b

b


Solve the right triangle

A= 28.00o, and c = 17.4 ft


Homework

Page 73 # 9-14


Definitions

Angle of Elevation: from point X to point Y (above X) is the acute angle formed by ray XY and a horizontal ray with endpoint X.

Angle of Depression: from point X to point Y (below) is the acute angle formed by ray XY and a horizontal ray with endpoint X.


Solving an Applied Trigonometry Problem

Step 1 Draw a sketch, and label it with the given information. Label the quantity

to be found with a variable.

Step 2 Use the sketch to write an equation relating the given quantities to the variable.

Step 3 Solve the equation, and check that your answer makes sense.


Example: Application

The length of the shadow of a tree 22.02 m tall is 28.34 m. Find the angle of elevation of the sun.

Draw a sketch.

The angle of elevation of the sun is 37.85.

22.02 m

28.34 mB

1

22.02tan

28.3422.02

tan 37.8528.34

B

B


2.5

Further Applications of Right Triangles


Bearing

Other applications of right triangles involve bearing, an important idea in navigation.


Example

An airplane leave the airport flying at a bearing of N 32E for 200 miles and lands. How far east of its starting point is the plane?

The airplane is approximately 106 miles east of its starting point.

sin 32oe

200

e 200sin 32o

e 106

e

200

32º


Example: Using Trigonometry to Measure a Distance A method that surveyors use to determine a small

distance d between two points P and Q is called the subtense bar method. The subtense bar with length b is centered at Q and situated perpendicular to the line of sight between P and Q. Angle is measured, then the distance d can be determined.

a) Find d when = and b = 2.0000 cm b) Angle usually cannot be measured more accurately

than to the nearest 1 second. How much change would there be in the value of d if were measured 1 second larger?

1 23'12"


Example: Using Trigonometry to Measure a Distance continued

Let b = 2, change to decimal degrees.

b) Since is 1 second larger, use 1.386944.

The difference is .0170 cm.

2

cot2

cot2 2

b

d

bd

1 23'12" 1.386667

2 1.386667co 8t 2.6341

2m

2

c

d

2 1.38694482co .61

26t

27 m c

d


Example: Solving a Problem Involving Angles of Elevation Sean wants to know the height of a Ferris wheel.

From a given point on the ground, he finds the angle of elevation to the top of the Ferris wheel is 42.3 . He then moves back 75 ft. From the second point, the angle of elevation to the top of the Ferris wheel is 25.4 . Find the height of the Ferris wheel.


Example: Solving a Problem Involving Angles of Elevation continued The figure shows two

unknowns: x and h. Since nothing is given about

the length of the hypotenuse, of either triangle, use a ratio that does not involve the hypotenuse, (the tangent ratio).

In triangle ABC,

In triangle BCD,

xC

B

h

DA 75 ft

42.3 25.4

tan 42.3 or tan 4 2.3 .h

hx

x

tan 25.4 or (75 ) tan 25 5

.4 .7

xh

hx


Example: Solving a Problem Involving Angles of Elevation continued Since each expression equals h, the

expressions must be equal to each other.

tan 42.3 75 tan 25.4 tan 25.4

tan 42.3 tan 25.4 75 tan 25.4

(tan 42.3 tan 25.4 ) 75 tan 25.4

75 tan 25.4

tan 42.3 tan 25

tan 42.3 (75

.4

) tan 25.4x x

x x

x x

x

x

Solve for x.

Distributive Property

Factor out x.

Get x-terms on one side.

Divide by the coefficient of x.


We saw above that Substituting for x.

tan 42.3 = .9099299 and tan 25.4 = .4748349. So, tan 42.3 - tan 25.4 = .9099299 - .4748349 = .435095 and

The height of the Ferris wheel is approximately 74.48 ft.

75 tan 25.4tan 42.3 .

tan 42.3 tan 25.4h

Example: Solving a Problem Involving Angles of Elevation continued

tan 42.3 .h x

75 .4748349 .435095

.9099299 74.47796

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