Gibilisco S. Everyday Math Demystified: A Self-Teaching Guide

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except  ¼ 908 (p/2 rad) and  ¼ 2708 (3p/2 rad):

sec

  tan

 ¼ 1

tan

  sec

 ¼1

USE CRUTCHES!

Trigonometry is a branch of mathematics with extensive applications

in science, engineering, architecture – even in art. You should not be shy

about using a calculator and making sketches to help yourself solve prob-

lems. So what if they’re ‘‘crutches’’? As long as you get the correct answer,

you can use any help you need. (Just be sure you keep a fresh, spare battery

around for your calculator if it’s battery powered.)

PROBLEM 12-7

Use a drawing of the unit circle to help show why it is true that sin

 þ

cos

 ¼ 1 for angles  greater than 08 and less than 908. (Hint: a right triangle

is involved.)

SOLUTION 12-7

Figure 12-10 shows a drawing of the unit circle, with  deﬁned counterclock-

wise between the x axis and a ray emanating from the origin. When the angle

is greater than 08 but less than 908, a right triangle is formed, with a segment

of the ray as the hypotenuse. The length of this segment is equal to the

radius of the unit circle. This radius, by deﬁnition, is 1 unit. According to

the Pythagorean theorem for right triangles, the square of the length of the

hypotenuse is equal to the sum of the squares of the lengths of the other

Fig. 12-10. Illustration for Problem 12-7.

CHAPTER 12 A Taste of Trigonometry 291

two sides. It is easy to see from Fig. 12-9 that the lengths of these other two

sides are sin  and cos . Therefore:

ðsin Þ

þðsin Þ

¼ 1

which is the same as saying that sin

 þ cos

 ¼ 1.

PROBLEM 12-8

Use another drawing of the unit circle to help show why it is true

that sin

 þ cos

 ¼ 1 for angles  greater than 2708 and less than 3608.

(Hint: this range can be considered, in directional terms, as equivalent to

the range of angles greater than 908 and less than 08.)

SOLUTION 12-8

Figure 12-11 shows how this can be done. Draw a mirror image of Fig. 12-10,

with the angle  deﬁned clockwise instead of counterclockwise. Again we have

a right triangle. This triangle, like all right triangles, obeys the Pythagorean

theorem.

Quiz

Refer to the text in this chapter if necessary. A good score is eight correct.

Answers are in the back of the book.

1. Suppose you stand on a ﬂat, level surface on a sunny morning. Your

shadow is exactly four times as long as you are tall. What is the angle

Fig. 12-11. Illustration for Problem 12-8.

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292

of the sun above the horizon, to the nearest degree?

(a) 148

(b) 458

(d) More information is needed to answer this question.

2. Suppose you stand on a ﬂat, level surface on a sunny morning. The sun is

at an angle of 508 above the horizon. You are exactly 160 centimeters

(cm) tall. How long is your shadow, to the nearest centimeter?

(a) 122 cm

(b) 134 cm

(d) More information is needed to answer this question.

3. Imagine that you stand some distance away from a tall building with a

ﬂat roof. The earth in the vicinity of the building is ﬂat and level. The

angle between the horizon and the top of the building, as measured

from a point at your feet, is 70.008. The straight-line distance between

your feet and the nearest point at the top of the building is 200 meters

(m). How tall is the building, to the nearest meter?

(a) 115 m

(b) 188 m

(d) More information is needed to answer this question.

4. One arc second is the equivalent of

(a) 1/60 of an arc minute

(b) 1/3600 of an arc minute

(d) 1/(2p) of an arc minute

5. Approximately how many degrees are there in 1 rad?

(a) 32.78

(b) 458

(d) None of the above.

6. Suppose an almanac lists the latitude of your town as 358 15

This is equivalent to

(a) 35.158 N

(b) 35.258 N

(d) None of the above.

CHAPTER 12 A Taste of Trigonometry 293

7. What is the angle, in radians, representing revolution by 3/4 of a

circle in a clockwise direction?

(a) 3p/4

(b) 3p/4

(d) 3p/2

8. The circumference of the earth is approximately 40,000 kilometers

(km). What is the angular separation between two points 111 km

apart, measured over the shortest possible path on the surface?

Neglect irregularities in the terrain. Assume the earth is a perfect

sphere.

(a) 1 rad

(b) 57.38

(d) 18

9. Suppose you live in a remote area and decide to install an outdoor

antenna for receiving stereo FM broadcasts. You buy a support

tower that is 20.00 meters (m) high. The installation guide recom-

mends guying the tower from a point halfway up, and also from

the top. The guide also recommends that all guy wires be sloped at

angles no greater than 458 with respect to the horizontal. Suppose

you install two sets of guy wires, one set from a point halfway up

and the other set from the top. All guys are fastened to three anchors

at ground level. Each guy anchor is at just the right distance from the

base of the tower to make the top set of guy wires slant at precisely

45.008. At what angle, to the nearest tenth of a degree and relative to

the horizontal, does each of the lower-level guy wires slant? Assume

the ground in the vicinity of the tower is ﬂat and level.

(a) 22.58

(b) 26.68

(d) More information is needed to answer this question.

10. What is the largest possible value for the tangent of an angle?

(a) 1

(b) 908

(d) None of the above.

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Test: Part 3

Do not refer to the text when taking this test. You may draw diagrams or use

a calculator if necessary. A good score is at least 30 correct. Answers are in

the back of the book. It’s best to have a friend check your score the ﬁrst time,

so you won’t memorize the answers if you want to take the test again.

1. Imagine that you are on a long, straight, level stretch of two-lane high-

way. A set of utility poles, all 20 m high, 40 m to the right of the center

line (as you see them while driving), and 70 m from each other, is

set up alongside the road. A perfectly taut, straight wire runs along

the tops of the poles. Which of the following statements (a), (b), (c),

or (d), if any, is true?

(a) The line of the wire is parallel to the highway center line.

(b) The line of the wire is skewed with respect to the highway center line.

indeﬁnitely, they would intersect at some point far in front of you.

(d) If the line of the wire and the highway center line were extended

indeﬁnitely, they would intersect at some point far behind you.

(e) None of the above statements (a), (b), (c), or (d) is true.

295

2. In a right triangle, the hypotenuse

(a) is always opposite the smallest angle

(b) is always opposite the 308 angle

(d) is always opposite the 1208 angle

(e) is always opposite the largest angle

3. Suppose you are an ant on a ﬂat, level ﬁeld, and you are looking at

a broadcast tower. Your companion, another ant, looks up at the

tower and says, ‘‘We are only 0.001 m tall, and that building is

100,000 times as tall as we are.’’ To this you reply, ‘‘The top of that

tower is at an angle of about 458 with respect to the horizon.’’ Your

companion, not to be outwitted, correctly observes:

(a) ‘‘If that is true, then we are roughly 10 m from the bottom of the

tower.’’

(b) ‘‘If that is true, then we are roughly 71 m from the bottom of the

tower.’’

(d) ‘‘If that is true, then we are roughly 710 m from the bottom of the

tower.’’

(e) ‘‘That tells us nothing about our distance from the tower.’’

4. Imagine a city laid out in square blocks on a level expanse of prairie,

with ‘‘streets’’ running east–west and ‘‘avenues’’ running north–south.

The distances between adjacent intersections are all exactly 200 m.

You stop at Gas Station Alpha and fuel up the car, and then you

ask the attendant how to get to Junk Market Beta. The attendant

says, ‘‘Go 5 blocks north, then turn left and go 12 blocks west.’’

What is the straight-line distance between Gas Station Alpha and

Junk Market Beta?

(a) It is impossible to answer this without more information.

(b) 1.3 km

(d) 3.4 km

(e) 6.0 km

5. An American style pro football ﬁeld is shaped like a rectangle, with

sidelines that are perpendicular to the goal lines. Imagine that the

two sidelines are extended indeﬁnitely in both directions. The resulting

pair of lines

(a) would intersect at a distant point

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296

(b) would intersect at two distant points

(d) would be parallel

(e) would be perpendicular

6. An American style pro football ﬁeld is shaped like a rectangle, with

yard lines that are all parallel to the goal lines. Fill in the blank to

make the following sentence true: ‘‘The yard lines on a football ﬁeld

are all ________________ with respect to each other.’’

(a) perpendicular

(b) skewed

(d) parallel

(e) orthogonal

7. In the polar coordinate plane, the direction can be expressed in

(a) meters

(b) degrees

(d) cubic units

(e) linear units

8. Refer to Fig. Test 3-1. Suppose lines L, M, and N are all straight, and

they all lie in the same plane. Suppose line M intersects line N at point

P, and line L intersects line N at point Q. Further suppose that lines L

and M are parallel. Let the angles of intersection, and their measures,

be denoted as p

through p

and q

through q

, as shown. Which of the

following statements (a), (b), (c), or (d) is not necessarily true?

(a) p

¼ q

(b) q

¼ q

Fig. Test 3-1. Illustration for Part Three Test Questions 8 through 11.

Test: Part 3 297

¼ p

(d) q

¼ q

(e) All of the statements (a) through (d) are true.

9. Refer to Fig. Test 3-1. Suppose lines L, M, and N are all straight, and

they all lie in the same plane. Suppose line M intersects line N at

point P, and line L intersects line N at point Q. Further suppose

that lines L and M are parallel. Let the angles of intersection, and

their measures, be denoted as p

through p

and q

through q

,as

shown. Angles p

and q

are called

(a) alternate interior angles

(b) alternate exterior angles

(d) horizontal angles

(e) corresponding angles

10. Refer to Fig. Test 3-1. Suppose lines L, M, and N are all straight, and

they all lie in the same plane. Suppose line M intersects line N at

point P, and line L intersects line N at point Q. Further suppose

that lines L and M are parallel. Let the angles of intersection,

and their measures, be denoted as p

through p

and q

through q

as shown. Angles p

and p

are called

(a) alternate interior angles

(b) alternate exterior angles

(d) horizontal angles

(e) corresponding angles

11. Refer to Fig. Test 3-1. Suppose lines L, M, and N are all straight, and

they all lie in the same plane. Suppose line M intersects line N at

point P, and line L intersects line N at point Q. Let the angles

of intersection, and their measures, be denoted as p

through p

and q

through q

, as shown. Suppose that p

¼ q

. From this we

can surmise that

(a) lines L and M are skew

(b) lines L and M are perpendicular

(d) lines L and M intersect to the left of points P and Q

(e) lines L and M intersect to the right of points P and Q

12. Imagine a city laid out in square blocks on a level expanse of

prairie, with ‘‘streets’’ running east–west and ‘‘avenues’’ running

north–south. The distances between adjacent intersections are all

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298

exactly 200 m. You stop at Restaurant Gamma and have supper, and

on the way out you ask the cashier how to get to Motel Delta. The

cashier says, ‘‘Go 8 blocks east, then turn right and go 6 blocks

south.’’ What is the distance you must drive to get to the motel?

(a) It is impossible to answer this without more information.

(b) 1.0 km

(d) 2.0 km

(e) 2.8 km

13. In Fig. Test 3-2, what is the distance between the labeled points

(3, 4) and (0, 0)?

(a) 3 units

(b) 4 units

(d) 7 units

(e) It is impossible to determine this without more information.

14. In Fig. Test 3-2, what is the slope of the heavy, slanted line?

(a) 4/3

(b) 3/4

(d) 4/3

(e) It is impossible to determine this without more information.

15. In Fig. Test 3-2, what is the equation of the heavy, slanted line?

(a) (4/3)x þ y ¼ 0

(b) (3/4)x þ y ¼ 0

Fig. Test 3-2. Illustration for Part Three Test Questions 13 through 16.

Test: Part 3 299

(d) (4/3)x þ y ¼ 0

(e) It is impossible to determine this without more information.

16. In Fig. Test 3-2, suppose L is a line perpendicular to the heavy,

slanted line. What is the equation of L?

(a) (4/3)x þ y ¼ 0

(b) (3/4)x þ y ¼ 0

(d) (4/3)x þ y ¼ 0

(e) It is impossible to determine this without more information.

17. Suppose you live in a house that is shaped like a rectangular prism.

It is exactly 15 m long, 20 m wide, and 5 m high, with a ﬂat roof.

There are two stories and no basement. Neglecting the volume

taken up by the interior walls, ﬂoors, ceilings, furniture, and other

objects, how much living room, in terms of volume, do you have?

(a) 75 m

(b) 100 m

(d) 600 m

(e) 1500 m

18. Suppose you live in a house that is shaped like a rectangular prism. It

is exactly 15 m long, 20 m wide, and 5 m high, with a ﬂat roof. There

are two stories and no basement. Neglecting the space taken up by

the interior walls, furniture, and other objects, how much living

room, in terms of ﬂoor area, do you have?

(a) 75 m

(b) 100 m

(d) 600 m

(e) 1500 m

19. How many seconds of arc are there in a complete circle?

(a) 2p

(b) 360

(d) 1,296,000

(e) It is impossible to answer this without more information.

20. Imagine a city laid out on a level expanse of prairie, with ‘‘streets’’

that are concentric circles spaced 200 m apart and ‘‘avenues’’

that go radially outward from the city center at angles that are all

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300