slide1
Download
Skip this Video
Download Presentation
Geometry Trigonometric Ratios

Loading in 2 Seconds...

play fullscreen
1 / 32

Geometry Trigonometric Ratios - PowerPoint PPT Presentation


  • 108 Views
  • Uploaded on

Geometry Trigonometric Ratios. Warm Up. Find the geometric mean of each pair of a number. 1) 3 and 27 2) 6 and 24 3) 8 and 32. Trigonometric Ratios.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Geometry Trigonometric Ratios' - lynnea


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

Geometry

Trigonometric

Ratios

CONFIDENTIAL

slide2

Warm Up

Find the geometric mean of each pair of a number.

1) 3 and 27 2) 6 and 24 3) 8 and 32

CONFIDENTIAL

slide3

Trigonometric Ratios

By the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with that same acute angle measure. So ∆ABC~∆DEF~∆XYZ, and BC/AC = EF/DF = YZ/XZ. These are trigonometric radios. A trigonometric ratio is a ratio of two sides of a right triangle.

32˚

32˚

32˚

CONFIDENTIAL

slide4

Trigonometric Ratios

DEFINITION

SYMBOLS

DIAGRAM

The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse.

B

c

a

A

The cosine of an angle is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse.

C

b

Next page 

CONFIDENTIAL

slide5

DEFINITION

SYMBOLS

DIAGRAM

The tangent of an angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle.

B

c

a

A

C

b

CONFIDENTIAL

slide6

Trigonometric Ratios

Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.

R

12

5

T

S

13

sin R

sin R = 12/13 0.92

cos R

cos R = 5/13 0.38

tan R

tan R = 5/12 0.42

~

~

~

~

~

~

CONFIDENTIAL

slide7

Now you try!

1) Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.

a. cos A b. tan B c. sin B

CONFIDENTIAL

slide8

Trigonometric Ratios in special Right Triangles

Use a special right to write sin 60˚ as a fraction.

Draw and label a 30˚ -60˚ -90˚∆.

CONFIDENTIAL

slide9

Now you try!

2) Use a special right triangle to write tan 45˚ as a fraction.

CONFIDENTIAL

slide10

Calculating Trigonometric Ratios

Use your calculator to find each trigonometric ratio. Round the nearest hundredth.

A ) cos 76˚ B) sin 8˚ C) tan 82˚

cos76˚= 0.24

sin 8˚= 0.14

tan 82˚=7.12

CONFIDENTIAL

slide11

Now you try!

3) Use your calculator to find each trigonometric ratio.

Round to the nearest hundredth.

a. tan 11˚ b. sin 62˚ c. cos 30˚

CONFIDENTIAL

slide12

The hypotenuse is always the longest side of a right triangle. So the denominator of a sine or cosine ratio is always greater than the numerator. Therefore the sine and cosine of an acute angle are always positive numbers less than 1. Since the tangent of an acute angle is the ratio of the lengths of the legs, it can have any value greater than 0.

CONFIDENTIAL

slide13

Trigonometric Ratios to Find Lengths

Find each length. Round to the nearest hundredth.

A) AB

AB is adjacent to the given angle, /A. You are given BC, which is opposite /A. Since the adjacent and opposite legs are involved, use a tangent ratio.

Write a trigonometric ratio.

Substitute the given values.

Multiply both sides by AB and divide by tan 41˚.

Simplify the expression.

Next page 

CONFIDENTIAL

slide14

B) MP

MP is opposite the given angle, /N. You are given NP, which is the hypotenuse. Since the opposite side and hypotenuse are involved, use a sine radio.

Write a trigonometric radio.

Substitute the given values.

Multiply both sides by 8.7 Simplify the expression.

Next page 

CONFIDENTIAL

slide15

C) YZ

YZ is the hypotenuse. You are given XZ, which is adjacent to the given angle, /Z. Since the adjacent side and hypotenuse are involved, use a cosine ratio.

Write a trigonometric ratio.

Substitute the given values.

Multiply both sides by YZ and divide by cos 38˚.

Simplify the expression.

CONFIDENTIAL

slide16

NOW YOU TRY!

4) Find each length. Round to the nearest hundredth

CONFIDENTIAL

slide17

Problem Solving Application

A contractor is building a wheelchair ramp for a doorway that is 1.2 ft above the ground. To meet ADA guidelines, the ramp will make an angle of 4.8˚ with the ground. To the nearest hundredth of a foot, what is the horizontal distance covered by the ramp?

Make a sketch. The answer is BC.

Write a trigonometric ratio.

Substitute the given values.

Multiply both sides by BC and divide by cos 4.8˚.

Simplify the expression.

CONFIDENTIAL

slide18

NOW YOU TRY!

5) Find AC, the length of the ramp in Example 5. to the nearest hundredth of a foot

CONFIDENTIAL

slide20

Assessment

Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.

1)sin C 2) tan A 3) cos A

CONFIDENTIAL

slide21

Use a special right triangle to write each trigonometric ratio as a fraction.

4) cos 60˚ 5) tan 30˚ 6) sin 45˚

CONFIDENTIAL

slide22

Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.

7) tan 67˚ 8) sin 23˚

CONFIDENTIAL

slide24

Let’s review

Trigonometric Ratios

By the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with that same acute angle measure. So ∆ABC~∆DEF~∆CYZ, and BC/AC = EF/DF = YZ/XZ. These are trigonometric radios. A trigonometric ratio is a ratio of two sides of a right triangle.

32˚

32˚

32˚

CONFIDENTIAL

slide25

Trigonometric Ratios

DEFINITION

SYMBOLS

DIAGRAM

The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse.

B

c

a

A

The cosine of an angle is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse.

C

b

Next page 

CONFIDENTIAL

slide26

DEFINITION

SYMBOLS

DIAGRAM

The tangent of an angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle.

B

c

a

A

C

b

CONFIDENTIAL

slide27

Calculating Trigonometric Ratios

Use your calculator to find each trigonometric ratio. Round the nearest hundredth.

A cos 76˚ B sin 8˚ C tan 82˚

CONFIDENTIAL

slide28

The hypotenuse is always the longest side of a right triangle. So the denominator of a sine or cosine ratio is always greater than the numerator. Therefore the sine and cosine of an acute angle are always positive numbers less than 1. Since the tangent of an acute angle is the ratio of the lengths of the legs, it can have any value greater than 0.

CONFIDENTIAL

slide29

Trigonometric Ratios to Find Lengths

Find each length. Round to the nearest hundredth.

A. AB

AB is adjacent to the given angle, /A. You are given BC, which is opposite /A. Since the adjacent and opposite legs are involved, use a tangent ratio.

Write a trigonometric ratio.

Substitute the given values.

Multiply both sides by AB and divide by tan 41˚.

Simplify the expression.

CONFIDENTIAL

slide30

Problem Solving Application

A contractor is building a wheelchair ramp for a doorway that is 1.2 ft above the ground. To meet ADA guidelines, the ramp will make an angle of 4.8˚ with the ground. To the nearest hundredth of a foot, what is the horizontal distance covered by the ramp?

Make a sketch. The answer is BC.

Write a trigonometric ratio.

Substitute the given values.

Multiply both sides by BC and divide by cos 4.8˚.

Simplify the expression.

CONFIDENTIAL

slide31

Trigonometric Ratios in special Right Triangles

Use a special right to write sin 60˚ as a fraction.

Draw and label a 30˚ -60˚ -90˚∆.

CONFIDENTIAL

ad