copyright © 2011 pearson education, inc. slide 8.7-1


Chapter 8: Trigonometric Functions and Applications

8.1 Angles, Arcs, and Their Measures

8.2 The Unit Circle and Its Functions

8.3 Graphs of the Sine and Cosine Functions

8.4 Graphs of the Other Circular Functions

8.5 Functions of Angles and Fundamental Identities

8.6 Evaluating Trigonometric Functions

8.7 Applications of Right Triangles

8.8 Harmonic Motion


8.7 Applications of Right Triangles

• Significant Digits– Represents the actual measurement.– Most values of trigonometric functions and

virtually all measurements are approximations.


8.7 Solving a Right Triangle Given an Angle and a Side

Example Solve the right triangle ABC, with A = 34º 30 and

c = 12.7 inches.

SolutionAngle B = 90º – A = 89º 60 – 34º 30

= 55º 30.

Use given information to find b.

inches 19.70334sin7.127.12

0334sin

sin

aa

ca

A

inches 5.100334cos7.127.12

0334cos bb


8.7 Solving a Right Triangle Given Two Sides

Example Solve right triangle ABC if a = 29.43 centimeters and c = 53.58 centimeters.

Solution Draw a sketch showing the given information.

Using the inverse sine functionon a calculator, we find A 33.32º.

B = 90º – 33.32º 56.68º

Using the Pythagorean theorem,

58.5343.29

sin A

s.centimeter 77.4443.2958.53 222

222

bb

acb


8.7 Angles of Elevation or Depression

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 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.


Solving an Applied Trigonometry Problem

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

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

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

8.7 Angles of Elevation or Depression


8.7 Solving a Problem Involving Angle of Elevation

Example The length of the shadow of a building 34.09 meters tall is 3.62 meters. Find the elevation of the sun.

The angle of elevation of the sun is 42.18o.

1

34.09tan37.62

34.09so tan 42.18 .37.62

B

B


8.7 Bearing

There are two methods for expressing bearing. When a single angle is given, it is understood that the bearing is measure in a clockwise direction from the north.


8.7 Solving a Problem Involving Bearing(First Method)

Example Radar stations A and B are on an east-west line, with A west of B, 3.70 km apart. Station A detects a plane at C, on a bearing of 61.0o. Station B simultaneously detects the same plane, on a bearing of 331.0o. Find the distance from A to C.


8.7 Solving a Problem Involving Bearing(First Method)

Solution Draw a sketch. Since a line drawn due north is perpendicular to an east-west line, right angles are formed at A and B, so angles CAB and CBA can be found. Angle C is a right angle. Find distance b by using the cosine function.

cos29.0 so3.70

3.70cos29.0 3.24 km

b

b


8.7 Bearing

The second method for expressing bearing starts with a north-south line and uses an acute angle to show the direction, either east or west, from this line.

Either N or S always come first, followed by an acute angle, and then E or W.


8.7 Solving a Problem Involving Bearing(Second Method)

Example The bearing from A to C is S 52o E. The bearing from A to B is N 84o E. The bearing from B to C is S 38o W. A plane flying at 250 mph takes 2.4 hours to go from A to B. Find the distance from A to C.



Solution Make a sketch. First draw the two bearings from point A. Then choose a point B on the bearing N 84o E from A, and draw the bearing to C. Point C will be located where the bearing lines from A and B intersect.

The distance from A to B is 250(2.4) = 600 miles.



Solution (continued)

To find b, the distance from A to C, use the sine function.

sin46

sin46600

600sin46 so

430 miles

bcb

b

b


8.7 Calculating the Distance to a Star

• In 1838, Friedrich Bessel determined the distance to a star called 61 Cygni using a parallax method that relied on the measurement of very small angles.

• You observe parallax when you ride in a car and see a nearby object apparently move backward with respect to more distance objects.

• As the Earth revolved around the sun, the observed parallax of 61 Cygni is 0.0000811º.



Example One of the nearest stars is Alpha Centauri, which has a parallax of 0.000212º.

(a) Calculate the distance to Alpha Centauri if the Earth-Sun distance is 93,000,000 miles.

(b) A light-year is defined to be the distance that light travels in 1 year and equals about 5.9 trillion miles. Find the distance to Alpha Centauri in light-years.



Solution(a) Let d be the distance between Earth and Alpha

Centauri. From the figure on slide 8-46,

(b) This distance equals

miles. 1051.2000212.sin

000,000,93sin

000,000,93or

000,000,93sin

13

d

dd

years.-light 3.4109.51051.2

12

13



Example Francisco needs to know the height of

a tree. From a given point on the ground, he finds

that the angle of elevation to the top of the tree is

36.7º. He then moves back 50 feet. From the second

point, the angle of elevation is 22.2º. Find the height

of the tree.



Analytic Solution There are two unknowns, the distance xand h, the height of the tree.

In triangle ABC,

In triangle BCD,

Each expression equals h, so the expressions must be equal.

.7.36tanor7.36tan xhxh

.2.22tan)50(or50

2.22tan xhx

h

2.22tan7.36tan2.22tan50

2.22tan502.22tan7.36tan

2.22tan2.22tan507.36tan

2.22tan)50(7.36tan

x

xx

xx

xx



We saw above that h = x tan 36.7º. Substituting for x,

Graphing Calculator Solution Superimpose the figure on the coordinate axes with D at the origin.

feet. 457.36tan2.22tan7.36tan

2.22tan50

h

Line DB has m = tan 22.2º with y-intercept 0. So the equation of line DB is y = tan 22.2º x.

Similarly for line AB, using the point-slope form of a line, we get the equation y = [tan 36.7º](x – 50).



Plot the lines DB and AB on the graphing calculator and find the point of intersection.

Rounding the information at the bottom of the screen, we see that h 45 feet.

Line DB: y = tan 22.2º x

Line AB: y = [tan 36.7º](x – 50).

copyright © 2011 pearson education, inc. slide 8.7-1

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