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6-1: Right-Triangle Trigonometry

© 2007 Roy L. Gover (www.mrgover.com)

Learning Goals:

•Define the six trig ratios of an acute angle•Evaluate trig ratios using triangles, a calculator and special angles

•Introduce trigonometry and angle measurement

Important IdeaTrigonometry, which means triangle measurement was developed by the Greeks in the 2nd century BC. It was originally used only in astronomy, navigation and surveying. It is now used to model periodic behavior such as sound waves and planetary orbits.

DefinitionAngle: the figure formed by two rays with a common endpoint

Definition

Initial Side

Terminal Side

Vertex Angle A

is in standard position

A x

y

43

1

Definition

x

yThe Cartesian Plane is divided into quadrants as follows:

2

Important Idea

Angles may be measured in degrees where 1 degree (°) is 1/360 of a circle, 90° is ¼ of a circle, 180° is ½ of a circle, 270° is ¾ of a circle and 360° is a full circle. A 90° angle is also called a right angle.

Example90°

180°

270°

0° or 360°

Try This

What is the measure of this angle?

a. 0°b. 45°c. 90°d. 120°e. 180°

DefinitionFractional parts of a degree can be written in decimal form or Degree-Minute –Second (DMS) form. A minute is 1/60 of a degree. A second is 1/60 of a minute.How many seconds are in each degree?

Example

Write 29°40’20” in decimal degrees accurate to 3 decimal places.

Symbol for minute

Symbol for second

Important IdeaTo convert from DMS to decimal, write the decimal expression 29°40’20” as:40 20

2960 3600

in your calculator.

Try This

77.399°

Write 77°23’56’’ in decimal form.

0°

90°

180°

270°

Try This

185.751°

Write 185°45’3’’ in decimal form.

0°

90°

180°

270°

Try This

319.541°

Write 319°32’28’’ in decimal degrees accurate to 3 decimal places.

0°

90°

180°

270°

ExampleWrite 37.576° in DMS form.

Procedure:1. Write the

decimal part .576° as .576 x 60=34.56’2. Write the decimal part of minutes as .56 x 60=33.6’’3. Round 33.6’’ to 34’’ for total 37°34’34’’

Try ThisWrite 185.651° in DMS form.

0°

90°

180°

270°185°39’4’’

Try ThisWrite 85.259° in DMS form.

0°

90°

180°

270°85°15’32’’

Definition

A

B

Ca

b

cD

E

F

d

e

f

m A m D If thena d

c f b e

c f a d

b e& &

Similar Triangles

Important IdeaIf 2 right triangles have equal angles, the corresponding ratios of their sides must be the same no matter the size of the triangles. This fact is the basis for trigonometry.

Example

A

B

Ca

b

c

D

E

F

d

e

f

30m A m D If and

2, 4a c and 3d then

?f

Try This

A B

C

ab

c D

E

F

d

e

f

60m A m D If and

2, 4c b and 6e then

?f

12

Definition

The hypotenuseis the sideopposite the 90° angle and is the longest side. The other 2 sides are legs.

Definition

The opposite sideis the leg opposite the given angle

AC

DefinitionThe adjacent sideis the leg next to the given angle (not the hypotenuse).

AC

Important IdeaRight triangles come in all sizes, shapes and orientations.

DefinitionFor a given acute angle in a right triangle:

The sine of written as is the ratio

sin

sin oppositehypotenu

se(see p.416 of your text):

Mem

oriz

e


The cosine of written ascos

cos adjacenthypotenu

se

is the ratio:

(see p.416 of your text):

Mem

oriz

e


The tangent of written as

tan

tan opposite adjacent

is the ratio:


Mem

oriz

e

opposite


The cosecant of written as

csc

1csc

sin

hypotenu

se

is the ratio:


Mem

oriz

e

adjacent


The secant of written assec

1sec

cos

hypotenu

se

is the ratio:


Mem

oriz

e

opposite


The cotangent of written as

cot

1cot

tan

adjacent

is the ratio:


Mem

oriz

e

Example

13

5

12

Evaluate the 6 trig ratios of the angle

Try This

35

4

Evaluate the 6 trig ratios of the angle

4sin

5

3cos

5 4

tan3

5csc

4 5

sec3

3cot

4

Try This

13

5

12Evaluate the 6 trig ratios of the angle

5sin

13

12cos

13 5

tan12

13csc

5 13

sec12

12

cot5

ExampleUsing your calculator, evaluate the 6 trig ratios of 33° Be sure that mode is set to degrees

Try ThisUsing your calculator, evaluate the 6 trig ratios of 117.25°

sin117.25 .889 cos117.25 .458 tan117.25 1.942

csc117.25 1.125sec117.25 2.184cot117.25 .515

Try ThisUsing your calculator, evaluatecos12 15'30''

cos12 15'30'' cos12.258 .977

Important Idea

1csc

sin

1sec

cos

Since your calculator does not have a sec, csc or cot key, you must find the reciprocal of cos, sin or tan.

1cot

tan

DefinitionThe special angles are:

•30°

•60°

•45°

Important Idea

These angles are special because they have exact value trig functions.

Consider the first two special angles in degrees...

30°

60°

Long Side

Short

Sid

e

Hypotenus

e

Analysis

Important Idea

In a 30°-60°-90° right triangle, the short side is opposite the 30° angle, the long side is opposite the 60° angle, and the hypotenuse is opposite the 90° angle.

Lon

g Sid

e

Hypotenuse

Short Side…orientation does not change the relationships between sides and angles

60°

30°

Important Idea

Lon

g Sid

e

Hypotenuse

Short Side

…orientation does not change the relationships between sides and angles

30°

60°

Important Idea

Important IdeaIn a 30°,60°,90° triangle:•the short side is one-half the hypotenuse.

•the long side is times the short side.

3Memoriz

e

Try ThisFind the length of the missing sides: 30

°

60°4

8

4 3

Try ThisFind the length of the missing sides:

10

5

5 330°

Try This

Find the length of the missing sides:

5

5 3

3

10 3

3

60°

Try ThisFind the length of the missing sides:

60°4

8

4 3

30°60°45°

45°

45°

AnalysisConsider the last special angle:

Hypot

enus

e…orientation does not change the relationships between sides and angles

45°

45°

Important Idea

Hypotenuse…orientation does not change the relationships between sides and angles

45°

45°

Important Idea

…and sides opposite equal angles are equal...

x

x

and by the2x

pythagorean theorem, the hypotenuse is...

45°

45°

Important Idea

Important IdeaIn a 45°,45°,90° triangle:•The legs of the triangle are equal.

•the hypotenuse is times the length of the leg.

Memoriz

e2

Try ThisFind the length of the missing sides

2

22 2

45°

Try ThisFind the length of the missing sides

2

2

2

45°

45°

ExampleFind the exact value of the 6 trig functions of 30°. This is not a calculator problem.

30°

Important Idea

When you know the lengths of the 3 sides of a right triangle, you can evaluate any of the 6 trig functions.

ExampleFind the exact value of the 6 trig functions of 45°. This is not a calculator problem.

45°

Try ThisFind the exact value of the 6 trig functions of 60°. Do not use a calculator.

60°

Solution

60°1

23

3sin 60

2

1cos60

2

tan 60 3

Solution

60°1

23

2 3csc60

3

sec60 2

3cot 60

3

Lesson CloseWe will use the information in this lesson to solve right triangle problems in the next lesson. Right triangle problems are used in real-world applications such as indirect measurement, surveying and navigation.

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Technology

definition angle

right angle

given angle

given acute angle

b e c f

opposite definition

b c ab c d e f d e f

verte x angle