Geometry HR Date: 4/16/2013 - Morgan Park High School...8.5 ≈ 0.4706 7.5 8.5 ≈ 0.8824 4 7.5 ≈ Trig ratios are often expressed as decimal approximations. Ex. 2: Finding Trig Ratios—Find

Geometry HR Date: 4/16/2013

Question: How do we measure the immeasurable?

Obj: SWBAT apply properties of 45-45-90 and 30-60-90

triangles.

Bell Ringer: 5 min check 1-4

HW Requests: HR pg 567 #1-12

In class: Red WB pg 101, 102

• HW:Red WB Skills

• Practice 8.4, pg 103

Announcements:

Quiz Wednesday

Geometry 4/16/13

Obj: SWBAT use trig ratios Question: How do we measure the immeasurable?

Agenda

• Bell Ringer: pg 559 #41, 48

• Special Assignments

• Homework Requests:HR 4th 6th period, Red WB

101,IB 567 #16-27

• Homework: Red WB Skills

• Practice 8.4, pg 103

• Announcements:

• Quiz Wednesday

Ex. 1: Finding Trig Ratios Large Small

15

817

A

B

C

7.5

48.5

A

B

C

sin A = opposite

hypotenuse

cosA = adjacent

hypotenuse

tanA = opposite

adjacent

8

17 ≈ 0.4706

15

17 ≈ 0.8824

8

15 ≈ 0.5333

4

8.5 ≈ 0.4706

7.5

8.5 ≈ 0.8824

4

7.5 ≈ 0.5333

Trig ratios are often expressed as decimal approximations.

Ex. 1: Finding Trig Ratios Large Small

15

817

A

B

C

7.5

48.5

A

B

C

sin A = opposite

hypotenuse

cosA = adjacent

hypotenuse

tanA = opposite

adjacent

8

17 ≈ 0.4706

15

17 ≈ 0.8824

8

15 ≈ 0.5333

4

8.5 ≈ 0.4706

7.5

8.5 ≈ 0.8824

4

7.5 ≈ 0.5333

Trig ratios are often expressed as decimal approximations.

Ex. 2: Finding Trig Ratios—Find the sine, the

cosine, and the tangent of the indicated angle.

R

sin S = opposite

hypotenuse

cosS = adjacent

hypotenuse

tanS = opposite

adjacent

12

13 ≈ 0.9231

5

13 ≈ 0.3846

12

5 ≈ 2.4

adjacent

opposite12

13 hypotenuse5

R

T S

Trigonometry is… • A branch of geometry used first by the

Egyptians and Babylonians (Iraq)

• Used extensively is astronomy and building

• Based on ratios between angles in RIGHT

Triangles

Why is this useful?

• Imagine being an Ancient Egyptian. They

had no calculators, no computers. They

could use one angle and a pyramid’s side

length to find the other side lengths.


cosine, and the tangent of 45

45

sin 45= opposite

hypotenuse

adjacent

hypotenuse

opposite

adjacent

1

hypotenuse1

√2

cos 45=

tan 45=

1

√2 =

√2

2 ≈ 0.7071

1

√2 =

√2

2 ≈ 0.7071

1

1 = 1

Begin by sketching a 45-45-90 triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. It follows that the length of the hypotenuse is √2.

45



45

sin 45= opposite

hypotenuse

adjacent

hypotenuse

opposite

adjacent

1

hypotenuse1

√2

cos 45=

tan 45=

1

√2 =

√2

2 ≈ 0.7071

1

√2 =

√2

2 ≈ 0.7071

1

1 = 1

Begin by sketching a 45-45-90 triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. It follows that the length of the hypotenuse is √2.

45

2

1



30

sin 30= opposite

hypotenuse

adjacent

hypotenuse

opposite

adjacent

√3

cos 30=

tan 30=

√3

2 ≈ 0.8660

1

2 = 0.5

√3

3 ≈ 0.5774

Begin by sketching a 30-60-90 triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. It follows that the length of the longer leg is √3 and the length of the hypotenuse is 2.

30

√3

1 =

2

1



30

sin 30= opposite

hypotenuse

adjacent

hypotenuse

opposite

adjacent

√3

cos 30=

tan 30=

√3

2 ≈ 0.8660

1

2 = 0.5

√3

3 ≈ 0.5774

Begin by sketching a 30-60-90 triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. It follows that the length of the longer leg is √3 and the length of the hypotenuse is 2.

30

√3

1 =

!IMPORTANT!

To solve trig functions in the calculator,

make sure to set your MODE to DEGREES

• Directions:

Press MODE, arrow down to Radian

Arrow over to Degrees

Press ENTER

Example:

Find the missing side lengths.

• We have one angle (30°) and the hypotenuse.

• Which ratios can we use to find the other sides?

Trig Ratios

For an ∠X Inverse Function

To find trig ratio To find the ∠X

Sin ∠X = Opp Sin-1 Opp = ∠X

Hyp Hyp

Cos ∠X = Adj Cos-1 Adj = ∠X

Hyp Hyp

Tan ∠X =Opp Tan-1 Opp = ∠X

Adj Adj

ReferenceTriangles

n

n

2n

45

45 1

1

2

45

45

This is our reference triangle for the 45-45-90.

n 2n

30

60

3n

1 2

30

60

3This is our reference triangle for the 30-60-90.

The Trigonometric Functions

SINE

COSINE

TANGENT

Pronounced “theta”

Greek Letter q

Represents an unknown angle

Pronounced “alpha”

Greek Letter α


Pronounced “Beta”

Greek Letter β


Finding Trig Ratios

• A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan respectively.

Trigonometric Ratios • Let ∆ABC be a right

triangle. The sine,

the cosine, and the

tangent of the acute

angle A are defined

as follows.

ac

bside adjacent to angle A

Side

opposite

angle A

hypotenuse

A

B

C

sin A = Side opposite A

hypotenuse

= a

c

cos A = Side adjacent to A

hypotenuse

= b

c

tan A = Side opposite A

Side adjacent to A

= a

b

q

opposite hypotenuse

SinOpp

Hyp

adjacent

CosAdj

Hyp

TanOpp

Adj

hypotenuse opposite

adjacent

C B

A

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

adjacent

SOHCAHTOA

It is important to note WHICH angle you are talking

about when you find the value of the trig function.

a

b

c

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

4

5

sin = Use a mnemonic and

figure out which sides

of the triangle you

need for sine.

h

o

5

3

tan =

a

o

3

4

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

4

5

Oh,

I'm

acute!

So

am I!

Geometry HR Date: 4/16/2013 - Morgan Park High School...8.5 ≈ 0.4706 7.5 8.5 ≈ 0.8824 4 7.5 ≈ Trig ratios are often expressed as decimal approximations. Ex. 2: Finding Trig Ratios—Find

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