No Slide Title

Transcript No Slide Title

Chapter 2

Acute Angles and Right Triangles

2.1

Trigonometric Functions of Acute Angles

Development of Right Triangle Definitions of B Trigonometric Functions c a  Let ABC represent a right triangle with right A b 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.

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 A c B b C a

Slide 2-3

Development of Right Triangle Definitions of Trigonometric Functions  Based on this diagram, each of the six trigonometric functions for angle A would be defined: sin A  a c c csc A  a cos A  b c sec A  c b B c tan A  a b cot A  b a A b C a

Slide 2-4

B  Right Triangle Definitions of hypotenuse c a side opposite A Trigonometric Functions A b C side adjacent t o A The same ratios could have been obtained without placing an acute angle in standard position by making the following definitions: sin A  side opposite A hypotenuse cos A  side adjacent t o A hypotenuse csc A  hypotenuse side opposite A sec A  hypotenuse side adjacent t o A tan A  side opposite A side adjacent t o A cot A  side adjacent t o A side opposite A  Standard

“Right Triangle Definitions”

of Trigonometric Functions ( MEMORIZE THESE!!!!!! )

May help to memorize " soh cah toa"

Slide 2-5

Example: Finding Trig Functions of Acute Angles

 Find the values of sin

, cos

, and tan

in the right triangle shown.

sin A  cos A  tan A  side opposite A hypotenuse  20 52  5 13 side adjacent t o A  hypotenuse side opposite A side adjacent t o A  48 52  12 13 20 48  5 12

52 48

Slide 2-6

Development of Cofunction Identities

A  sin Given any right triangle, ABC, how does the measure of B compare with A?

c B = 90

 A a A  a c  cos B  cos  90

 A B  csc A  c a  sec B  sec  90

tan A  a b  cot B  cot  90

C b

Slide 2-7

Cofunction Identities

  By similar reasoning other cofunction identities can be verified: For any acute angle

, sin

tan

= cos(90   = cot(90  

A A

) ) csc

= sec(90  

) cos

A =

sin(90  

) sec

= csc(90  

) cot

= tan(90  

)

MEMORIZE THESE!

!

Slide 2-8

Example: Write Functions in Terms of Cofunctions

 Write each function in terms of its cofunction.

 a) cos 38  =  b) sec 78  = sin (90   38  ) = csc (90   78  ) =

sin 52



csc 12

 The cofunction identities can be described as : The function of an angle is equal to the cofunction of its complement .

Slide 2-9

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

Slide 2-10

Example: Solving Equations

 Assuming that all angles are acute angles, find one solution for the equation: cot(4   8 )  tan(2   4 ).

cot  4   8

  cot  90

  2   4

  Angles don' t have 4  4  to be equal for   8

  cotangents 90 90

o o

   2  2  to be equal,  4

  4

6   6 8  

   86

but it is one way they can be equal

Slide 2-11

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

Slide 2-12

Example: Comparing Function Values

 Tell whether the statement is

true

sin 31  > sin 29 Referring to  drawing, sin which or

false

31 is o 

31 and sin

bigger,

31 or

r y

29 ?

o 

sin 31

o 

sin 29

is TRUE!

o r x

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

Slide 2-13

Equilateral Triangles

 Triangles that have three equal side lengths are equilateral  Equilateral triangles also have three equal angles each measuring 60 o  All equilateral triangles are similar (corresponding sides are proportional)

2  1 2  2 2 60 o

2 2 3 30 o 2

2  1  4 60 o 60 o 60 o

2  3 2 1

 3

Slide 2-14

Using 30-60-90 Triangle to Find Exact Trigonometric Function Values  30-60-90 Triangle  Find each of these: cos 60 0  1 2 tan 60 0  3 sin sec 30 30 0 0   1 2 2 3  2 3 3

MEMORIZE THIS!

Slide 2-15

Isosceles Right Triangles

  Right triangles that have two legs of equal length Also have two angles of measure 45 o  All such triangles are similar 1 45 o

45 o 1 2  1 2 

2 2 

Slide 2-16

Using 45-45-90 Triangle to Find Exact Trigonometric Function Values  45-45-90 Triangle  Find each of these: cos 45 0  1 2  2 2 tan 45 0  1 sin 45 0  1 2  2 2 sec 45 0  2

MEMORIZE THIS!

Slide 2-17

Function Values of Special Angles

Are used a lot in Trigonomet ry and can quickly be determined  sin  from memorized cos  tan  triangles cot  .

sec  csc  30  45  1 2 2 2 2 3 2 2 3 3 1 3 1 2 3 3 2 2 2 60  2 3 1 2 3 3 3 2 2 3 3 This chart can also be quickly completed by memorizing the first column and using identities .

Slide 2-18

Usefulness of Knowing Trigonometric Functions of Special Anlges: 30 o , 45 o , 60 o  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 30 o , 45 o and 60 o angles that are somehow related to trig function values of these angles

Slide 2-19

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

Slide 2-20

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

-axis. (Shown below in red)

' '

Slide 2-22

Example: Find the reference angle for each angle.

   218  Positive acute angle made by the terminal side of the angle and the

-axis is: 218   180  =

     1387  First find coterminal angle between 0 o and 360 o 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  360  – 307  =

 is:

Slide 2-23

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



 ,

  



  ,

' y

   

 ,

  



 ,

   

Slide 2-24

Comparison of Trigonometric Functions of Angles vs Functions of Reference Angles     Based on the observations on the previous slide:

Trigonometric functions of any angle

the

same

value will be

as trigonometric functions of

its

reference angle

except for

the

sign

of the answer The

sign quadrant

of the answer can be

determined by

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

Slide 2-25

Finding Trigonometric Function Values for Any Non-Acute Angle

 

Step

1  

Step

3 

Step

4 If 

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  .

Find the reference angle 

Find the trigonometric function values for reference angle 

Determine 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 

Slide 2-26

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  =

60 0 1

Slide 2-27

  

Example Continued

2 60 0 1 30 0 3 Trig functions of any angle are equal to trig functions of its reference angle except that sign is determined from quadrant of angle 210 o is in quadrant III where only tangent and cotangent are positive Based on these observations, the six trig functions of 210 o are: sin 210 o   sin 30 o   1 2 csc 210 o   csc 30 o   2 cos 210 o   cos 30 o tan 210 o  tan 30 o    2 3 sec 210 o   sec 30 o   2 3 3 3 3 cot 210 o  cot 30 o  3

Slide 2-28

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 0 2 1 30 0  cos   240 0 cos      3 cos     1 2

Slide 2-29

Expressions Containing Powers of Trigonometric Functions    An expression such as: Has the meaning:

sin

 sin 2    2 Example: Using your memory regarding side ratios of 30-60-90 and 45-45-90 triangles, simplify:

sin

45

0 

tan

30

0   2 2  2   3 3  2  2 4  3 9  1 2  3 9  9 18  6 18  3 18  1 6

Slide 2-30

Example: Evaluating an Expression with Function Values of Special Angles

 Evaluate cos 120  + 2 sin 2 60   tan 2 30  .

  Individual trig function values before evaluating are: cos 120   1 2 , sin 60  2 3 , and tan 30  3 3 , Substituting into the expression: cos 120  + 2 sin 2 60   tan 2 30    1 2 2 + 2    2 3    2     2 3    3 9  2 3 3 3    2 Copyright © 2005 Pearson Education, Inc.

Slide 2-31

    Finding Unknown Special Angles that Have a Specific Trigonometric Function Value   0 0 , 360 0  cos    2 2 Use your knowledge of trigonometric function values of 30 o , 45 o 60 o angles* to find a reference angle that has the same absolute value as the specified function value and 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

Slide 2-32

    Example: Finding Angle Measures Given an Interval and a Function Value    cos    2 2 Which special angle has the same absolute 0 value cosine as this angle?

45 In which quadrants is cosine negative?

II and III Putting 45 o reference angles in quadrants II and III, gives which two angles as answers?

180 0  45 0  135 0 180 0  45 0  225 0

Slide 2-33

Homework

  2.2 Page 59 All: 1 – 6, 10 – 17, 25 – 32, 36 – 37, 48 – 53, 61- 66  MyMathLab Assignment 2.2 for practice  MyMathLab Homework Quiz 2.2 will be due for a grade on the date of our next class meeting

Slide 2-34

2.3

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

Slide 2-36

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 This is the

sec A

Slide 2-37



Example: Finding Function Values with a Calculator

sin 38 24  Convert 38  24  to decimal degrees and use sin key.

 cot 68 .

4832 o Find tan of the angle and use reciprocal key 38 24   38 24 6 0 sin 38 24  38.4

 si n 38.4

 .6211477

cot 68.4832   .3942492

Slide 2-38

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

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

is an appropriate number, then gives the measure of

sin  1

, cos  1

, or tan -1

angle* whose sine, cosine, or tangent is

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

Slide 2-39



Example: Using Inverse Trigonometric Functions to Find Angles

sin   .8535508

Using the degree mode and the inverse sine value .8535508 is 58.6 .

 We write the result as 1 sin .8535508

 58.6

Slide 2-40

  

Example: Using Inverse Trigonometric Functions to Find Angles continued

sec   2.486879

Use reciprocal identities to get: cos   1 2 .

486879  .

4021104 The result is:   66.289824

Slide 2-41

Homework

  2.3 Page 64 All: 5 – 29, 55 – 62  MyMathLab Assignment 2.3 for practice  MyMathLab Homework Quiz 2.3 will be due for a grade on the date of our next class meeting

Slide 2-42

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”

Slide 2-44

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: 4 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: 2 3 6

Slide 2-45

Significant Digits for Angles

 The following conventions are used in expressing accuracy of measurement (significant digits) in angle measurements Number of Significant Digits 2 3 4 5 Angle Measure to Nearest: Degree Ten minutes, or nearest tenth of a degree Minute, or nearest hundredth of a degree Tenth of a minute, or nearest thousandth of a degree Copyright © 2005 Pearson Education, Inc.

Slide 2-46

Calculations Involving Significant Digits

2 4  

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

Examples: 32 , 000 .

0072  4444444 .

4 according to calculator Significan t Digits?

32 , 000  4 , 400 , 000 two significan .

0072 t digits 2 3200 sin 5 42 .

458 0  2160 .

1586 according to calculator 3200 sin Significan 42 .

458 0 t Digits?

 2200 two significan t digits

Slide 2-47

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

Slide 2-48

Example: Solving a Right Triangle, Given an Angle and a Side

 Solve right triangle

ABC

, if

= 42  30' and

= 18.4.

 How would you find angle B?

= 90  42  30'

= 47  Which trig 30 ‘ = 47.5

 function relates A, a and c?

sin sin



42 .

5 0



18 .

4 18 .

4 sin 42 .

5 0 18 .

4  .

675590207   

a a

Which trig

12 .

4 

a C

function

b c =

18.4

42  30' relates A, b and c?

cos



b c

cos 42 .

5 18 .

4 cos 0 

18 .

4 42 .

5 0 

13 .

6 

b A

Slide 2-49

Example: Solving a Right Triangle Given Two Sides

 Solve right triangle

ABC

= 11.47 cm and

= 27.82 cm.

What trig function relates A and the two given sides?

sin sin



opposite hypotenuse A

 .

412293314  11 .

47 27 .

B a

= 11.47

c =

27.82

 sin  1 .

412293314

 24 .

35 0 (Note " significan How cos would

 11 27 .

you 47  .

82 .

find B?

412293314

 cos  1 .

412293314  65 .

65 0 t digits" )

C b

How would

2 

2  you

2 find b?

2  27.82

 25.35

2  11.47

Slide 2-50

Angles of “Elevation” and “Depression”

Some applicatio n problems involve " angle of elevation"  and " angle of depression " Angle of Elevation: from point

to point

(above

) is the acute angle formed by ray

and a horizontal ray with endpoint

 Angle of Depression: from point

to point

(below) is the acute angle formed by ray

and a horizontal ray with endpoint

Slide 2-51

Solving an Applied Trigonometry Problem 

Step 1



Step 2



Step 3

Draw a sketch, and label it with the given information. Label the quantity to be found with a variable.

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

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

Slide 2-52

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 26 o 40’. If her eyes are 5.30 ft above the ground, find the height of the pole.

 height of pole above horizontal

5 .

30 123 tan 123 tan 26 0 4 0  26 0 4 0  

123 

Height of pole from ground?

61 .

8  5 .

30  67 .

1 ft.

61 .

8 

Slide 2-53

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.

Equation?

 tan

 22.02

28.34

 tan  1 22.02

 37.85

28.34 m 28.34

The angle of elevation of the sun is 37.85

 .

22.02 m

Slide 2-54

Homework

  2.4 Page 72 All: 11 – 14, 21 – 28, 35 – 36, 41 – 44, 48 – 49  MyMathLab Assignment 2.4 for practice  MyMathLab Homework Quiz 2.4 will be due for a grade on the date of our next class meeting

Slide 2-55

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 0 o 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 90 o )

Slide 2-57

Describing Bearing Using First Method

  Note: All directions can be described as an angle in the interval: [ 0 o , 360 ) Show bearings: 32 o , 164 o , 229 o and 304 o N N N N 32 0 164 0 229 0 304 0

Slide 2-58

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

Slide 2-59

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 61 o , while station B simultaneously detects the same plane on a bearing of 331 o . N Find the distance from A to C.

N Right tria ngle is formed!* 61 0

29 0 C 90 0 61 0 Trig function relating cos

, 3.7

and an acute angle?

29 0 

3 .

7 3 .

7 cos 29 0 

A 3 .

with any triang 3 .

2 km 

le using Law of Cosines

Slide 2-60

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 30 o W would represent 30 o rotation to the west of a north line N 30 0  Example: S 45 o E would represent 45 o east of a south line rotation to the S 45 0

Slide 2-61

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?

Equation involving trig function?

 sin 32



200  200sin 32

 106 200 32 The airplane is approximately 106 miles west of its starting point.

Slide 2-62

Using Trigonometry to Measure a Distance

 A method that surveyors use to determine a small distance

between two points

and

is called the

subtense bar method

. The subtense bar with length

centered at

and situated perpendicular to the line of sight between

and

. Angle  is measured, then the distance

can be determined.

cot 

2 



2 cot  2 is

Slide 2-63

Example: Using Trigonometry to Measure a Distance

  Find

when  and

cot  = 2.0000 cm

2 



2 cot  2 Let

= 2, change  decimal degrees.

Significan

t Digits



82 .

634 cm :

 2 2 co t 1.386667

2  8 2.6341

Slide 2-64

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 42.3

o . He then moves back 75 ft. From the second point, the angle of elevation to the top of the Ferris wheel is 25.4

o . Find the height of the Ferris wheel.

Slide 2-65

Example: Solving a Problem Involving Angles of Elevation continued

    The figure shows two unknowns:

and

Use the two triangles, to write two trig function equations involving the two unknowns: In triangle

ABC

, tan 42.3



h x

In triangle

BCD

, 

tan 4 2.3 .

B h C x

42.3

25.4

75 ft

tan 25.4



7 5 

 (75 

) tan 25 .4 .

system of equations by substituti on.

Slide 2-66

Example: Solving a Problem Involving Angles of Elevation continued

 Since each expression equals

, the expressions must be equal to each other.

tan 42.3

 (75 

) tan 25.4

 75 tan 25.4



tan 25.4

tan 42.3

(tan 42.3



tan 25.4

 75 tan 25.4

 tan 25.4 )  75 tan 25.4

Resulting Equation Distributive Property Get

-terms on one side.

Factor out

 75 tan 25.4

tan 42.3

 tan 25 .4

Divide by the coefficient of

Slide 2-67



Example: Solving a Problem Involving Angles of Elevation continued

h h



tan 42.3 .

.  75 tan 25.4

tan 42.3

 tan 25.4

  tan 42.3 .

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

and    .435095

    .9099299

 74.

 The height of the Ferris wheel is approximately 74 ft.

Slide 2-68

Homework

  2.5 Page 81 All: 11 – 16, 23 – 28  MyMathLab Assignment 2.5 for practice  MyMathLab Homework Quiz 2.5 will be due for a grade on the date of our next class meeting

Slide 2-69

No Slide Title

Transcript No Slide Title

Chapter 2

Acute Angles and Right Triangles

2.1

Trigonometric Functions of Acute Angles

May help to memorize " soh cah toa"

Example: Finding Trig Functions of Acute Angles

Development of Cofunction Identities

Cofunction Identities

MEMORIZE THESE!

!

!

Example: Write Functions in Terms of Cofunctions

Example: Solving Equations

Example: Comparing Function Values

sin 31

sin 29

is TRUE!

Equilateral Triangles

MEMORIZE THIS!

Isosceles Right Triangles

MEMORIZE THIS!

Function Values of Special Angles

Homework

2.2

Trigonometric Functions of Non-Acute Angles

Reference Angles

Example: Find the reference angle for each angle.

Finding Trigonometric Function Values for Any Non-Acute Angle

Example Continued

Example: Finding Trig Function Values Using Reference Angles

sin

sin

45

tan

30

Example: Evaluating an Expression with Function Values of Special Angles

Homework

2.3

Function Values Using a Calculator

Function Values Using a Calculator

Example: Finding Function Values with a Calculator

Example: Using Inverse Trigonometric Functions to Find Angles

Example: Using Inverse Trigonometric Functions to Find Angles continued

Homework

2.4

Solving Right Triangles

Significant Digits

Significant Digits for Angles

Calculations Involving Significant Digits

Solving a Right Triangle

Example: Solving a Right Triangle, Given an Angle and a Side

Example: Solving a Right Triangle Given Two Sides

Angles of “Elevation” and “Depression”

Example: Application

Example: Application

Homework

2.5

Further Applications of Right Triangles

Describing Bearing Using First Method

Example

Example: Using Bearing

Using Trigonometry to Measure a Distance

Example: Using Trigonometry to Measure a Distance

Significan

t Digits

82 .

634 cm :

Example: Solving a Problem Involving Angles of Elevation

Example: Solving a Problem Involving Angles of Elevation continued

Example: Solving a Problem Involving Angles of Elevation continued

Example: Solving a Problem Involving Angles of Elevation continued

Homework

Directory