SECTION 14-5

• Applications of Right Triangles

Slide 14-5-1

APPLICATIONS OF RIGHT TRIANGLES

• Calculator Approximations for Function Values• Finding Angles Using Inverse Functions• Significant Digits• Solving Triangles• Applications

Slide 14-5-2

CALCULATOR APPROXIMATIONS FOR FUNCTION VALUES

Slide 14-5-3

Because calculators differ among makes and models, students should always consult their owner’s manual for specific information concerning their use.

When evaluating trigonometric functions of angles given in degrees, it is a common error to use the incorrect mode; remember that the calculator must be set in the degree (not radian) mode.

EXAMPLE: FINDING FUNCTION VALUES WITH A CALCULATOR

Slide 14-5-4

Use a calculator to approximate the value of each trigonometric function.

Solution

a) sin 54 12 b) sec32.662 c) tan( 321 )

a) sin 54 12 sin(54.20 ) .81106382 1

b) sec32.662 1.18783341cos32.662

c) tan( 321 ) .80978403

FINDING ANGLES USING INVERSE FUNCTIONS

Slide 14-5-5

We have used a calculator to find trigonometric function values of angles. This process can be reversed using inverse functions. Inverse functions are denoted by using –1 as a superscript. For example, the inverse of f is denoted f –1. For now we restrict our attention to angles in the interval The measure of an angle can be found from one of its trigonometric function values using inverse functions as show on the next slide.

0 90 .

EXAMPLE: USING INVERSE FUNCTIONS TO FIND ANGLES

Slide 14-5-6

Use a calculator to find a value for such that and satisfies each of the following.

0 90 .

a) sin .715332 b) sec 1.21112

Solution

a) 45.6704 1 1

b) cos .8256820133sec 1.21112

34.3423

Use sin-1 (2nd and then sin).

Use cos-1.

SIGNIFICANT DIGITS

Slide 14-5-7

A significant digit is a digit obtained by actual measurement. A number that represents the result of counting, a a number that results from theoretical work and is not the result of measurement, is an exact number.

Most values of trigonometric functions are approximations, and virtually all measurements are approximations. To perform calculations on such approximate numbers, follow the rules on the next slide.

CALCULATION WITH SIGNIFICANT DIGITS

Slide 14-5-8

For adding and subtracting, round the answer so that the last digit you keep is in the rightmost column in which all the numbers have significant digits.

For multiplying or dividing, round the answer to the least number of significant digits found in any of the given numbers.

For powers and roots, round the answer so that it has the same number of significant digits as the numbers whose power or root you are finding.

SIGNIFICANT DIGITS FOR ANGLES

Slide 14-5-9

Number of Significant Digits

Angle Measure to the Nearest:

2 Degree

3 Ten minutes, or nearest tenth of a degree

4 Minute, or nearest hundredth of a degree

5 Tenth of a minute, or nearest thousandth of a degree

SOLVING TRIANGLES

Slide 14-5-10

To solve a triangle means to find the measures of all the angles and all the sides of a triangle. In using trigonometry to solve triangles, a labeled sketch is useful.

A

B

C

a

b

c

EXAMPLE: SOLVING A RIGHT TRIANGLE GIVEN AN ANGLE AND A SIDE

Slide 14-5-11

A

B

C

a

b

14.8

Solve the triangle below.

40.3sin 40.314.8

a

(sin 40.3 )14.8 9.57a

cos 40.314.8

b

(cos 40.3 )14.8 11.3b

B = 90° – A

= 90° – 40.3° = 49.7°

Solution

EXAMPLE: SOLVING A RIGHT TRIANGLE GIVEN TWO SIDES

Slide 14-5-12

A

B

C

a

23.1

42.5

Solve the triangle below.

23.1cos

42.5A

57.1A 2 2 2a b c

B = 90° – 57.1° = 32.9°

Solution

2 2 223.1 42.5a 35.7a

APPLICATIONS

Slide 14-5-13

The 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 at X. The angle of elevation is always measured from the horizontal.

The angle of depression from point X to point Y (below X) is the acute angle formed by ray XY and a horizontal ray with endpoint at X.

Angle of elevation

X

Y

Horizontal

Angle of depression

X

Y

Horizontal

EXAMPLE: ANGLE OF ELEVATION

Slide 14-5-14

The length of the shadow of a building 35.28 meters tall is 40.45 meters. Find the angle of elevation of the sun.

35.28 m

40.45 m

35.28tan

40.45

41.09

The angle of elevation of the sun is 41.09°.

Solution