Trigonometric RatiosStudents will define and apply sine,

cosine, and tangent ratios to right triangles.

Discover the relationship of

the trigonometric ratios for

similar triangles.

Trigonometric RatiosExplain the relationship between the

trigonometric ratios of

complementary angles.Solve application

problems using the

trigonometric ratios.

Trigonometric Ratios in Right Triangles

Mr. Jack

Trigonometric Ratios are based on the Concept of Similar Triangles!

All 45º- 45º- 90º Triangles are Similar!

45 º

2

2

22

45 º

1

1

2

45 º

1

2

1

2

1

All 30º- 60º- 90º Triangles are Similar!

1

60º

30º

½

23

32

60º

30º

2

4

2

60º

30º

1

3

All 30º- 60º- 90º Triangles are Similar!

10 60º

30º

5

35

2 60º

30º1

3

160º

30º 21

23

hypotenuse

leg

leg

In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse

a

b

c

We’ll label them a, b, and c and the angles and . Trigonometric functions are defined by taking the ratios of sides of a right triangle.

First let’s look at the three basic functions.

SINECOSINE

TANGENT

They are abbreviated using their first 3 letters

c

a

hypotenuse

oppositesin

oppositec

b

hypotenuse

adjacentcos

adjacent

b

a

adjacent

oppositetan

The Trigonometric Functions

SINE

COSINE

TANGENT

SINE

Prounounced “sign”

Prounounced “co-sign”

COSINE

Prounounced “tan-gent”

TANGENT

Pronounced “theta”

Greek Letter

Represents an unknown angle

Pronounced “alpha”

Greek Letter α

Represents an unknown angle

Pronounced “Beta”

Greek Letter β

Represents an unknown angle

oppositehypotenuse

SinOpp

Hyp

adjacent

CosAdj

Hyp

TanOpp

Adj

hypotenuseopposite

adjacent

We could ask for the trig functions of the angle by using the definitions.

a

b

c

You MUST get them memorized. Here is a mnemonic to help you.

The sacred Jedi word:

SOHCAHTOA

c

b

hypotenuse

oppositesin

adjacentcos

hypotenuse

a

c opposite

tanadjacent

b

a

opposite

adjacent

SOHCAHTOA

It is important to note WHICH angle you are talking about when you find the value of the trig function.

a

bc

Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so

222 cba Let's choose: 222 5 43 3

45

sin = Use a mnemonic and figure out which sides of the triangle you need for sine.

h

o5

3

opposite

hypotenuse

tan =

a

o3

4

opposite

adjacent

Use a mnemonic and figure out which sides of the triangle you need for tangent.

You need to pay attention to which angle you want the trig function of so you know which side is opposite that angle and which side is adjacent to it. The hypotenuse will always be the longest side and will always be opposite the right angle.

This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle.

3

45

Oh, I'm

acute!

So am I!

We need a way to remember all of these ratios…

What is SohCahToa?

Is it in a tree, is it in a car, is it in the sky or is it from the deep blue sea ?

This is an example of a sentence using the word SohCahToa.

I kicked a chair in the middle of the night and my first thought was

I need to SohCahToa.

An example of an acronym for SohCahToa.Sevenold horsesCrawled a hill To our attic..

Old Hippie

Old Hippie

SomeOldHippieCameAHoppin’ThroughOurApartment

SOHCAHTOA

Old Hippie

Old Hippie

SinOppHypCosAdjHypTanOppAdj

Other ways to remember SOH CAH TOA1.Some Of Her Children Are Having Trouble Over Algebra. 2.Some Out-Houses Can Actually Have Totally Odorless Aromas. 3.She Offered Her Cat A Heaping Teaspoon Of Acid. 4.Soaring Over Haiti, Courageous Amelia Hit The Ocean And ... 5.Tom's Old Aunt Sat On Her Chair And Hollered. -- (from Ann Azevedo)

Other ways to remember SOH CAH TOA1.Stamp Out Homework Carefully, As Having Teachers Omit Assignments. 2.Some Old Horse Caught Another Horse Taking Oats Away. 3.Some Old Hippie Caught Another Hippie Tripping On Apples. 4.School! Oh How Can Anyone Have Trouble Over Academics.

A

Trigonometry Ratios

Tangent A =

opposite

adjacent

Sine A =

opposite

hypotenuse

Cosine A =

adjacent

hypotenuse

Soh Cah Toa

14º

24º

60.5º

46º 82º

1.9 cm

7.7 cm

14º

1.9

7.7

0.25 Tangent 14º

0.25

The Tangent of an angle is the ratio of the opposite side of a triangle to its adjacent side.

oppositeadjacent

hypotenuse

3.2 cm

7.2 cm24º

3.2

7.2

0.45 Tangent 24º

0.45

Tangent A =

opposite

adjacent

5.5 cm

5.3 cm

46º

5.5

5.3

1.04 Tangent 46º

1.04

Tangent A =

opposite

adjacent

6.7 cm

3.8 cm

60.5º

6.7

3.8

1.76

Tangent 60.5º

1.76

Tangent A =

opposite

adjacent

As an acute angle of a triangle approaches 90º, its tangent

becomes infinitely large

Tan 89.9º = 573

Tan 89.99º = 5,730

Tangent A =

opposite

adjacent

etc.

very large

very small

Since the sine and cosine functions alwayshave the hypotenuse as the denominator,

and since the hypotenuse is the longest side,these two functions will always be less than 1.

Sine A =

opposite

hypotenuse

Cosine A =

adjacent

hypotenuse

ASine 89º = .9998

Sine 89.9º = .999998

3.2 cm7.9 cm

24º

9.7

2.3

0.41 Sin 24º

0.41

Sin α = hypotenuse

opposite

5.5 cm

7.9 cm

46º

9.7

5.5

0.70 Cos 46º

0.70

Cosine β = hypotenuse

adjacent

A plane takes off from an airport an an angle of 18º and a speed of 240 mph. Continuing at this speed and angle,

what is the altitude of the plane after 1 minute?

18º

x

After 60 sec., at 240 mph, the plane has traveled 4 miles

4

18º

x4

opposite

hypotenuse

SohCahToaSine A =

opposite

hypotenuse Sine 18 =

x

4

0.3090 =

x

4

x = 1.236 milesor

6,526 feet

1

Soh

An explorer is standing 14.3 miles from the base of Mount Everest below its highest peak. His angle of

elevation to the peak is 21º. What is the number of feet from the base of Mount Everest to its peak?

21º14.3

x

Tan 21 =

x

14.30.3839 =

x

14.3

x = 5.49 miles = 29,000 feet

1

A swimmer sees the top of a lighthouse on the edge of shore at an 18º angle. The lighthouse is

150 feet high. What is the number of feet from theswimmer to the shore?

18º

150

Tan 18 =

x

150

x

0.3249 =

150

x

0.3249x = 150

0.3249 0.3249

X = 461.7 ft1

A dragon sits atop a castle 60 feet high. An archer stands 120 feet from the point on the ground directly

below the dragon. At what angle does the archer need to aim his arrow to slay the dragon?

x

60

120

Tan x =

60

120Tan x = 0.5

Tan-1(0.5) = 26.6º

Solving a Problem withthe Tangent Ratio

60º

53 ft

h = ?

We know the angle and the We know the angle and the side adjacent to 60º. We want to side adjacent to 60º. We want to know the opposite side. Use theknow the opposite side. Use thetangent ratio:tangent ratio:

ft 92353

531

3

5360tan

h

h

h

adj

opp

1

2 3

Why?

A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree?

50

71.5°

?

tan 71.5°

tan 71.5° 50

y

y = 50 (tan 71.5°)

y = 50 (2.98868)

149.4y ft

Example

Opp

Hyp

A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge?

200

x

Ex. 5

60°

cos 60°

x (cos 60°) = 200

x

X = 400 yards

Trigonometric Functions on a Rectangular Coordinate System

x

y

Pick a point on theterminal ray and drop a perpendicular to the x-axis.

ry

x

The adjacent side is xThe opposite side is yThe hypotenuse is labeled rThis is called a REFERENCE TRIANGLE.

y

x

x

yx

r

r

x

y

r

r

y

cottan

seccos

cscsin

Trigonometric Ratios may be found by:

45 º

1

1

2Using ratios of special trianglesUsing ratios of special triangles

145tan2

145cos

2

145sin

For angles other than 45º, 30º, 60º you will need to use a For angles other than 45º, 30º, 60º you will need to use a calculator. (Set it in Degree Mode for now.)calculator. (Set it in Degree Mode for now.)