Section 8.5Section 8.5

Tangent RatioTangent Ratio

What is What is TrigonometryTrigonometry??

The study of triangles and The study of triangles and their measurements.their measurements.

Three Trig. RatiosThree Trig. Ratios

1)1) SineSine2)2) CosineCosine3)3) TangentTangent

Why use this??????Why use this??????

35

y

x

5

A

C

B

What is the What is the tangent ratiotangent ratio??

ABBREVIATED FORMABBREVIATED FORM

leg opposite of tangent of

leg adjacent of

AA

A

opposite tan

adjacentA

oppositeleg

adjacent leg

A

5

1312

Y

ZX

1.1. Find tan Find tan X X and write as and write as a ratio.a ratio.

2.2. Find tanFind tan Y Y and write as and write as a ratio.a ratio.

opposite

adjacent

12

5

opposite tan

adjacentA

5

12

On you calculator.On you calculator.

On your calculator locate the On your calculator locate the sinsin, , coscos, and , and tantan buttons. buttons.

Then make sure your Then make sure your MODEMODE is is set to set to DEGREESDEGREES..

Use Use sin, cos, sin, cos, andand tan tan when you when you KNOW THE ANGLE KNOW THE ANGLE MEASUREMENTMEASUREMENT

On you calculator.On you calculator.

Notice Notice sinsin-1-1, , coscos-1-1, and , and tantan-1-1 above above the regular sin, cos, tan buttons.the regular sin, cos, tan buttons.

Those stand for inverse sin, Those stand for inverse sin, inverse cos, and inverse tan. inverse cos, and inverse tan.

Use these buttons with the 2Use these buttons with the 2ndnd button.button.

Use them when you Use them when you DON’T DON’T KNOW THE ANGLE.KNOW THE ANGLE.

Easy stuff – on you Easy stuff – on you calculator.calculator.

Find tangent of the following angles.Find tangent of the following angles.

3) tan 463) tan 46oo==

4) tan 784) tan 78oo==

5) tan 5) tan ? ? = 0.1944 = 0.1944

6) tan 6) tan ? ? = 6.3138 = 6.3138

1.035530314

4.704630109

11.00108712o

81.00006798o

216 81 97 z 297 z z

y

x

4

9H I

J

z

7) Find the measure of the 7) Find the measure of the missing angles and the missing missing angles and the missing

length.length.

97

2 2 24 9 z

9tan

4x 1 9

tan 66.03754

180 – 90 – 66.0375

= 23.9625

Find the missing Find the missing anglesangles..

8) 8) 8

tan15

A

17

8

B

A

C

15 (it’s a triple!)15

1 8tan

15

1 8tan 28.0725

15m A

180 – 90 – 28.0725

= 61.9275 =m∠B

Find the value of Find the value of xx..

27tan 40o

x

270.8390996

x

270.8390996 x x

x

0.8390996 27x 0.8390996 27

0.8390996 0.8390996

x

32.177x x

40

27

O

24

20

x

oA

B

C

Find tan Find tan A,A, mm∠∠B, B, tan tan BB

10) 10)

tan 24 0.44523o

180 90 24 66om B

tan 66 2.24604o

Road GradesRoad Grades

11) The grade of a road is the 11) The grade of a road is the ratioratio of its rise to its run and is of its rise to its run and is usually given as a percent. Find usually given as a percent. Find the angle that the road makes the angle that the road makes with the horizontal if its grade is with the horizontal if its grade is 4.5%.4.5%.

4.5 94.5%

100 200

Horizontal

xo

4.5%9

200

9tan

200x

1 9tan

200

1 9tan 2.57657 3

200o

PRACTICEPRACTICE

Page 306 – 307Page 306 – 307Classroom ExercisesClassroom Exercises#1 - 9 #1 - 9