Copyright © 2005 Pearson Education, Inc. Chapter 2 Acute Angles and Right Triangles.

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 2-3

Development of Right Triangle Definitions of Trigonometric Functions

Let ABC represent a right triangle with right angle at C and angles A and B as acute angles, with side “a” opposite A, side “b” opposite B and side “c” (hypotenuse) opposite C.

Place this triangle with either of the acute angles in standard position (in this example “A”):

Notice that (b,a) is a point on the

terminal side of A at a distance

“c” from the origin b

ab,

A

B

C

ac

bA

B

C

ac


Development of Right Triangle Definitions of Trigonometric Functions

Based on this diagram, each of the six trigonometric functions for angle A would be defined:

b

ab,

A

B

C

ac

a

bAcot

b

aAtan

b

cA sec

c

bA cos

a

cA csc

c

aAsin


Right Triangle Definitions ofTrigonometric Functions

The same ratios could have been obtained without placing an acute angle in standard position by making the following definitions:

Standard “Right Triangle Definitions” of Trigonometric Functions( MEMORIZE THESE!!!!!! )

bA

B

C

acA opposite side

A oadjacent t side

hypotenuse

A opposite side

A oadjacent t sideAcot

A oadjacent t side

A opposite sideAtan

A oadjacent t side

hypotenuseA sec

hypotenuse

A oadjacent t sideA cos

A opposite side

hypotenuseA csc

hypotenuse

A opposite sideAsin

toa"-cah-soh" memorize tohelpMay


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

Atan

A cos

Asin

A oadjacent t side

A opposite side

hypotenuse

A oadjacent t side

hypotenuse

A opposite side

12

5

48

2013

12

52

48

13

5

52

20


Development of Cofunction Identities

Given any right triangle, ABC, how does

the measure of B compare with A?

B = B C

A

bc

aAsin

c

a B

A90 o

cos A90 ocos

A csc a

c sec A90 osec

A tan b

a cot A90 ocot

B

B


Cofunction Identities

By similar reasoning other cofunction identities can be verified:

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)

!!THESE! MEMORIZE


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

terms of its cofunction.

a) cos 38 =

sin (90 38) =

sin 52

b) sec 78 =

csc (90 78) =

csc 12

.complement its of cofunction the toequal is anglean offunction The

:as described becan identities cofunction The


Solving Trigonometric Equations Using Cofunction Identities

Given a trigonometric equation that contains two trigonometric functions that are cofunctions, it may help to find solutions for unknowns by using a cofunction identity to convert to an equation containing only one trigonometric function as shown in the following example


Example: Solving Equations

Assuming that all angles are acute angles, find one solution for the equation:

cot(4 8 ) tan(2 4 ).

cot84cot o oo 4290

ooo 429084 equal becan way they one isit but equal, be tocotangentsfor equal be tohavet don' Angles

ooo 429084 oo 8686 o786 o13


Comparing the relative values of trigonometric functions

Sometimes it may be useful to determine the relative value between trigonometric functions of angles without knowing the exact value of either one

To do so, it often helps to draw a simple diagram of two right triangles each having the same hypotenuse and then to compare side ratios


Example: Comparing Function Values Tell whether the statement is true or false.

sin 31 > sin 29

Generalizing, in the interval from 0 to 90, as the angle increases, so does the sine of the angle

Similar diagrams and comparisons can be done for the other trig functions

o29o31 29y

31yrr

29x31x

r

y

r

y 29o31o 29sin and 31sin

? or bigger, is which drawing, toReferring 2931

r

y

r

y

TRUE! is 29sin 31sin oo


Equilateral Triangles

Triangles that have three equal side lengths are equilateral

Equilateral triangles also have three equal angles each measuring 60o

All equilateral triangles are similar (corresponding sides are proportional)

o60

2

2

2o60

o60 2o30

o60

1

h

3

3

41

21

2

2

222

h

h

h

h

3


Using 30-60-90 Triangle to Find Exact Trigonometric Function Values

30-60-90 Triangle Find each of these:

THIS! MEMORIZE

060tan

030sin

030sec

060cos2

1

3

2

1

3

2

3

32


Isosceles Right Triangles

Right triangles that have two legs of equal length Also have two angles of measure 45o

All such triangles are similar

2

1

1

o45

o45c

c

c

c

2

2

112

222


Using 45-45-90 Triangle to Find Exact Trigonometric Function Values

45-45-90 Triangle Find each of these:

THIS! MEMORIZE

045tan

045sin

045sec

045cos 2

1

2

2

2

1

2

2

1

2


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

. trianglesmemorized from determined be

quicklycan andry Trigonometin lot a used Are

.identities using andcolumn first thememorizing

by completedquickly be alsocan chart This


Usefulness of Knowing Trigonometric Functions of Special Anlges: 30o, 45o, 60o

The trigonometric function values derived from knowing the side ratios of the 30-60-90 and 45-45-90 triangles are “exact” numbers, not decimal approximations as could be obtained from using a calculator

You will often be asked to find exact trig function values for angles other than 30o, 45o and 60o angles that are somehow related to trig function values of these angles


Homework

2.1 Page 51 All: 1 – 14, 16 – 21, 23 – 26, 29 – 32, 35 – 42

MyMathLab Assignment 2.1 for practice

MyMathLab Homework Quiz 2.1 will be due for a grade on the date of our next class meeting


2.2

Trigonometric Functions of Non-Acute Angles


Reference Angles

A reference angle for an angle is the positive acute angle made by the terminal side of angle and the x-axis. (Shown below in red)

'

' '

'by indicated is for angle Reference


Example: Find the reference angle for each angle.

218 Positive acute angle made by

the terminal side of the angle and the x-axis is:

218 180 = 38

1387 First find coterminal angle

between 0o and 360o

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


Comparison of Trigonometric Functions of Angles vs Functions of Reference Angles

Each angle below has the same reference angle Choosing the same “r” for a point on the terminal

side of each (each circle same radius), you will notice from similar triangles that all “x” and “y” values are the same except for sign

yx ,

yx , yx ,

yx , '

' '

'


Comparison of Trigonometric Functions of Angles vs Functions of Reference Angles

Based on the observations on the previous slide: Trigonometric functions of any angle will be

the same value as trigonometric functions of its reference angle, except for the sign of the answer

The sign of the answer can be determined by quadrant of the angle

Also, we previously learned that the trigonometric functions of coterminal angles always have equal values


Finding Trigonometric Function Values for Any Non-Acute 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. (Hint: All students take calculus.) This gives the values of the trigonometric functions for angle .


Example: Finding Exact Trigonometric Function Values of a Non-Acute Angle

Find the exact values of the trigonometric functions for 210. (No Calculator!)

Reference angle:

210 – 180 = 30 Remember side ratios for

30-60-90 triangle. Corresponding sides:

2 ,3 ,1 12

3030

060


Example Continued

Trig functions of any angle are equal to trig functions of its reference angle except that sign is determined from quadrant of angle

210o is in quadrant III where only tangent and cotangent are positive

Based on these observations, the six trig functions of 210o are:

3 30cot210cot 3

330tan210tan

3

3230sec210 sec

2

330cos210cos

230csc210 csc 2

130sin210sin

oooo

oooo

oooo

12

3030

060


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

cos (240)

Coterminal angle between 0 and 360: 240 + 360 = 120

the reference angles is:

180 120 = 60 0240cos 0120cos

060cos2

1

12

3030

060


Expressions Containing Powers of Trigonometric Functions

An expression such as:

Has the meaning:

Example: Using your memory regarding side ratios of 30-60-90 and 45-45-90 triangles, simplify:

2sin

2sin

0202 30tan45sin

22

3

3

2

2 9

3

4

2

9

3

2

1

18

6

18

9

18

3

6

1


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

Individual trig function values before evaluating are:

Substituting into the expression:

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


Finding Unknown Special Angles that Have a Specific Trigonometric Function Value

Example: Find all values of in the interval given:

Use your knowledge of trigonometric function values of 30o, 45o and 60o angles* to find a reference angle that has the same absolute value as the specified function value

Use your knowledge of signs of trigonometric functions in various quadrants to find angles that have both the same absolute value and sign as the specified function value

*NOTE: Later we will learn to use calculators to solve equations that don’t necessarily have these special angles as reference angles

00 360 ,0

2

2cos


Example: Finding Angle Measures Given an Interval and a Function Value

Find all values of in the interval given:

Which special angle has the same absolute value cosine as this angle?

In which quadrants is cosine negative? Putting 45o reference angles in quadrants II and

III, gives which two angles as answers?

00 360 ,0

2

2cos

045III and II

000 13545180 000 22545180


Homework

2.2 Page 59 All: 1 – 6, 10 – 17, 25 – 32, 36 – 37, 48 – 53,

61- 66




2.3

Finding Trigonometric Function Values Using a Calculator


Function Values Using a Calculator

As previously mentioned, 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.

Also, angles measured in degrees, minutes and seconds must be converted to decimal degrees

Remember that most calculator values of trigonometric functions are approximations.


Function Values Using a Calculator

Sine, Cosine and Tangent of a specific angle may be found directly on the calculator by using the key labeled with that function

Cosecant, Secant and Cotangent of a specific angle may be found by first finding the corresponding reciprocal function value of the angle and then using the reciprocal key label x-1 or 1/x to get the desired function value

Example: To find sec A, find cos A, then use the reciprocal key to find:

This is the sec A value

Acos

1


Example: Finding Function Values with a Calculator

Convert 38 to decimal degrees and use sin key.

Find tan of the angle and use reciprocal key

cot 68.4832 .3942492

38 38

sin 38 24 si

38.4

38.4

2424

6

n

.6211477

0

sin 38 24

24

o4832.68cot


Finding Angle Measures When a Trigonometric Function of Angle is Known

When a trigonometric ratio is known, and the angle is unknown, inverse function keys on a calculator can be used to find an angle* that has that trigonometric ratio

Scientific calculators have three inverse functions each having an “apparent exponent” of -1 written above the function name. This use of the superscript -1 DOES NOT MEAN RECIPROCAL

If x is an appropriate number, then gives the measure of an angle* whose sine, cosine, or tangent is x.

* There are an infinite number of other angles, coterminal and other, that have the same trigonometric value

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 Find one value of given: Use reciprocal identities to get:

Now find using the inverse cosine function.

The result is:

66.289824

sec 2.486879

4021104.486879.2

1cos


Homework

2.3 Page 64 All: 5 – 29, 55 – 62




2.4

Solving Right Triangles


Measurements Associated with Applications of Trigonometric Functions

In practical applications of trigonometry, many of the numbers that are used are obtained from measurements

Such measurements many be obtained to varying degrees of accuracy

The manner in which a measured number is expressed should indicate the accuracy

This is accomplished by means of “significant digits”


Significant Digits

“Digits obtained from actual measurement” All digits used to express a number are

considered “significant” (an indication of accuracy) if the “number” includes a decimal The number of significant digits in 583.104 is: The number of significant digits in .0072 is:

When a decimal point is not included, then trailing zeros are not “significant” The number of significant digits in 32,000 is: The number of significant digits in 50,700 is: 3

2

46


Significant Digits for Angles

The following conventions are used in expressing accuracy of measurement (significant digits) in angle measurements

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


Calculations Involving Significant Digits

An answer is no more accurate than the least accurate number in the calculation

Examples:

calculator toaccording 4.44444440072.

000,32

digitst significan two000,400,40072.

000,32

calculator toaccording 1586.2160458.42sin3200 0

digitst significan two2200458.42sin3200 0

Digits?t Significan

2

4

Digits?t Significan

2 5


Solving a Right Triangle

To “solve” a right triangle is to find the measures of all the sides and angles of the triangle

A right triangle can be solved if either of the following is true: One side and one acute angle are known Any two sides are known


Example: Solving a Right Triangle, Given an Angle and a Side Solve right triangle ABC, if

A = 42 30' and c = 18.4. How would you find angle B?

B = 90 42 30'

B = 47 30‘ = 47.5 AC

B

c = 18.4

4230'

c

aA sin

4.185.42sin 0 a

a05.42sin4.18

a675590207.4.18

a4.12

c? and a A, relatesfunction Which trigc? and b A, relatesfunction Which trig

c

bA cos

4.185.42cos 0 b

b05.42cos4.18

b6.13

a

b


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

AC

B

c = 27.82a = 11.47

2 2 2

2 2 227.82 11.47

25.35

b c a

b

b

sides?given two theandA relatesfunction What trig

B? findyou wouldHow)digits"t significan" (Note

b? findyou wouldHow

b

82.27

47.11sin

hypotenuse

oppositeA

412293314.sin A

412293314.sin 1A035.24A

412293314.82.27

47.11cos B

01 65.65412293314.cos B


Angles of “Elevation” and “Depression”

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.

"depression of angle" and

elevation" of angle" involve problemsn applicatio Some


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

Shelly McCarthy stands 123 ft from the base of a flagpole, and the angle of elevation to the top of the pole is 26o40’. If her eyes are 5.30 ft above the ground, find the height of the pole.

30.5123

horizontal above pole ofheight xx

1230426tan 0 x

x0426tan123 0

x8.61

ground? from pole ofHeight

ft. 1.6730.58.61


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

Equation?


Homework

2.4 Page 72 All: 11 – 14, 21 – 28, 35 – 36, 41 – 44, 48 – 49




2.5

Further Applications of Right Triangles


Describing Direction by Bearing(First Method)

Many applications of trigonometry involve “direction” from one point to another

Directions may be described in terms of “bearing” and there are two widely used methods

The first method designates north as being 0o and all other directions are described in terms of clockwise rotation from north (in this context the angle is considered “positive”, so east would be bearing 90o)


Describing Bearing Using First Method

Note: All directions can be described as an angle in the interval: [ 0o, 360 )

Show bearings: 32o, 164o, 229o and 304o

NN N N

032

0164

02290304


Hints on Solving Problems Using Bearing

Draw a fairly accurate figure showing the situation described in the problem

Look at the figure to see if there is a triangular relationship involving the unknown and a trigonometric function

Write an equation and solve the problem


Example

Radar stations A and B are on an east-west line 3.7 km apart. Station A detects a plane at C on a bearing of 61o, while station B simultaneously detects the same plane on a bearing of 331o. Find the distance from A to C.

061

0331

090

A B

C

7.3

NN

d029 061

formed!* is ngleRight triaangle? acutean and 3.7 , relatingfunction Trig d

7.329cos 0 d

d029cos7.3

dkm 2.3Cosines of Law using leany triang with done beCan *


Describing Direction by Bearing(Second Method)

The second method of defining bearing is to indicate degrees of rotation east or west of a north line or east or west of a south line

Example: N 30o W would represent 30o rotation to the west of a north line

Example: S 45o E would represent 45o rotation to the east of a south line

N

S

030

045


Example: Using Bearing

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

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

sin32200

200sin32

106

e

e

e

e

200

32

function? triginvolvingEquation


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.

2

cot2

cot2 2

b

d

bd


Example: Using Trigonometry to Measure a Distance Find d when =

and b = 2.0000 cm

Let b = 2, change to decimal degrees.

2

cot2

cot2 2

b

d

bd

1 23'12" 1.386667

2 1.386667co 8t 2.6341

2m

2

c

d

1 23'12"

cm 634.82

:Digitst Significan

d


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

He doesn’t know his distance from the base of the wheel, but, from a given point on the ground, he finds the angle of elevation to the top of the Ferris wheel is 42.3o . He then moves back 75 ft. From the second point, the angle of elevation to the top of the Ferris wheel is 25.4o. Find the height of the Ferris wheel.


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

unknowns: x and h. Use the two triangles, to

write two trig function equations involving the two unknowns:

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

on.substitutiby equations of system thisSolve


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

Resulting Equation

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 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 .4748349.9099299 74.

.435095


Homework

2.5 Page 81 All: 11 – 16, 23 – 28



Copyright © 2005 Pearson Education, Inc. Chapter 2 Acute Angles and Right Triangles.

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Transcript of Copyright © 2005 Pearson Education, Inc. Chapter 2 Acute Angles and Right Triangles.