Slide 5.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide 5.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Right Triangle Trigonometry

Learn the definitions of the trigonometric functions of acute angles.Learn to evaluate trigonometric functions of acute angles.Learn the values of the trigonometric functions for the special angles 30º, 45º, and 60º.Learn to use right triangle trigonometry in applications.

SECTION 5.2

1

2

3

4


TRIGONOMETRIC RATIOS AND FUNCTIONS

a = length of the side opposite b = length of the side adjacent to c = length of the hypotenuse


RIGHT TRIANGLE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS OF AN

ACUTE ANGLE

sin opposite

hypotenusea

c

cos adjacent

hypotenuseb

c

tan opposite

adjacenta

b

csc hypotenuse

oppositec

a

sec hypotenuse

oppositec

b

cot adjacent

oppositeb

a


EXAMPLE 1Finding the Values of Trigonometric Functions

Find the exact values for the six trigonometric functions of the angle in the figure.

Solution

a2 b2 c2

3 2 7 2 c2

9 7 c2

16 c2

4 c


EXAMPLE 1Finding the Values of Trigonometric Functions

Solution continued

sin opp

hyp

3

4

cos adj

hyp

7

4

tan opp

adj

3

7

3 7

7

csc hyp

opp

4

3

sec hyp

adj

4

7

4 7

7

cot adj

opp

7

3

Now, with c4, a 3 and b 7 we have


TRIGONOMETRIC FUNCTION VALUES OF SOME COMMON ANGLES


TRIGONOMETRIC FUNCTION VALUES OF SOME COMMON ANGLES

6

30º1

23

2

3

32

2 3

33

deg

sin cos tan csc sec tanradians

4

45º2

2

2

21 2 2 1

3

60º1

23

2

3

32

2 3

33


COMPLEMENTARY ANGLESThe value of any trigonometric function of an acute angle is equal to the cofunction of the complement of . This is true whether is measured in degrees or in radians.

If is measured in radians, replace 90º with2

.

in degreessin cos 90º cos sin 90º tan cot 90º cot tan 90º sec csc 90º csc sec 90º


EXAMPLE 5Finding Trigonometric Function Values of a Complementary Angle

a. Given that cot 68º 0.4040, find tan 22º .

Solution

a. tan 22º cot 90º 22º cot 68º 0.4040

b. Given that cos 72º0.3090, findsin18º .

b. sin18º cos 90º 18º cos 72º 0.3090


EXAMPLE 7 Measuring the Height of Mount Kilimanjaro

A surveyor wants to measure the height of Mount Kilimanjaro by using the known height of a nearby mountain. The nearby location is at an altitude of 8720 feet, the distance between that location and Mount Kilimanjaro’s peak is 4.9941 miles, and the angle of elevation from the lower location is 23.75º . See the figure on the next slide. Use this information to find the approximate height of Mount Kilimanjaro.





SolutionThe sum of the side length h and the location height of 8720 feet gives the approximate height of Mount Kilimanjaro. Let h be measured in miles. Use the definition of sin , for = 23.75º.

sin opposite

hypotenuse

h

4.9941h 4.9941 sinh 4.9941 sin 23.75º

h 2.0014



Solution continued

1 mile = 5280 feet

2.0114 miles 2.0114 5280 feet

10,620 feet

Thus, the height of Mount Kilimanjaro

10,620 8720 19, 340 feet


EXAMPLE 8 Finding the Width of a River

To find the width of a river, a surveyor sights straight across the river from a point A on her side to a point B on the opposite side. See the figure on the next slide. She then walks 200 feet upstream to a point C. The angle that the line of sight from point C to point B makes with the river bank is 58º. How wide is the river?





The river is about 320 feet wide at the point A.

w

200tan 58º

w 200 tan 58º

w 320.07 feet

A, B, and C are the vertices of a right triangle with acute angle 58º. w is the width of the river.

Solution

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Slide 5.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley