PreCalculus 4/10/13

Obj: Midterm Review

Agenda

1. Bell Ringer: None

2. #35, 72 Parking lot 37, 39, 41

3. Homework Requests: Few minutes on Worksheet

4. Exit Ticket: In Class Exam Review

Homework:

Study for Midterm Exam

Announcements:

30th Week Exam 4/11

Exit Ticket:

Pg 368 #6, 10 also find the measure of the

angle.

Pg 368 #31, 33, 35,39, 49, 51

Pg 358 #4, 6, 14, 22, 26, 30,

Pg 346 #27, 32,33, 63, 75

Pg 331 #18, 37

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

The Trigonometric (trig) ratios: FUNCTION

INVERSE FUNCTION

Also true are…

Sample keystrokes Careful about Deg or

Rad Setting Exit Ticket pg 369 #30-40

evens Sample keystroke

sequences

Sample calculator display Rounded

Approximation

74

74

0.961262695 0.9613

0.275637355 0.2756

3.487414444 3.4874

sin

sin

ENTER

74

74

COS

COS

ENTER

74

74

TAN

TAN

ENTER

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 the swimmer to the shore?

18º

150

Tan 18 =

x

150

x

0.3249 =

150

x

0.3249x = 150

0.3249 0.3249

X = 461.7 ft 1

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

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

Exit Ticket WS 2, 4, 10, 12, 18, 20 For 18 find values of all trig functions

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?

21º 14.3

x

Tan 21 =

x

14.30.3839 =

x

14.3

x = 5.49 miles

= 29,000 feet

1

The Trigonometric Functions

SINE

COSINE

TANGENT

Pronounced “theta”

Greek Letter q

Represents an unknown angle

Pronounced “alpha”

Greek Letter α

Represents an unknown angle

Pronounced “Beta”

Greek Letter β

Represents an unknown angle

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!

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

S

sin S = opposite

hypotenuse

cosS = adjacent

hypotenuse

tanS = opposite

adjacent

5

13 ≈ 0.3846

12

13 ≈ 0.9231

5

12 ≈ 0.4167

adjacent

opposite

12

13 hypotenuse5

R

T S

Ex. 2: Finding Trig Ratios

S

sin S = opposite

hypotenuse

cosS = adjacent

hypotenuse

tanS = opposite

adjacent

5

13 ≈ 0.3846

12

13 ≈ 0.9231

5

12 ≈ 0.4167

adjacent

opposite

12

13 hypotenuse5

R

T S

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

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

Ex: 5 Using a Calculator

• You can use a calculator to approximate the

sine, cosine, and the tangent of 74. Make

sure that your calculator is in degree mode.

The table shows some sample keystroke

sequences accepted by most calculators.

3.2 cm

7.2 cm 24º

3.2

7.2

0.45 Tangent 24º

0.45

Tangent A =

opposite

adjacent

3.2 cm 7.9 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

x 4

opposite

hypotenuse

SohCahToa

Sine A =

opposite

hypotenuseSine 18 =

x

4

0.3090 =

x

4

x = 1.236 miles

or

6,526 feet

1

Soh

Solving a Problem with

the Tangent Ratio

60º

53 ft

h = ?

We know the angle and the

side adjacent to 60º. We want to

know the opposite side. Use the

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

Ex.

Opp

Hyp

Notes: • If you look back at Examples 1-5, you

will notice that the sine or the cosine of an acute triangles is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg or a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.

Ex

A 100 degree sector cut from a circular disc has a

length of 7 cm. To the nearest cm., what is the

radius of the circle? What is the area of the sector?

S = r ɵ

Must be in radians

Just convert degrees to

radians

ɵ

Ans. 4 cm

Ans.13.95 sq. cm

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.

Seven

old

horses

Crawled

a

hill

To

our

attic..

Old Hippie

Some Old Hippie Came A Hoppin’ Through Our Apartment

SOHCAHTOA

Old Hippie

Sin Opp Hyp Cos Adj Hyp Tan Opp Adj

Other ways to remember SOH CAH TOA

1. 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 TOA

1. 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.

opposite adjacent

hypotenuse

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 always have 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

A Sine 89º = .9998

Sine 89.9º = .999998

Ex. 3: Finding Trig Ratios—Find the sine, the

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. From Theorem 9.8 on page 551, it follows that the length of the hypotenuse is √2. 45

2

1

Ex. 4: Finding Trig Ratios—Find the sine, the

cosine, and the tangent of 30

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. From Theorem 9.9, on page 551, it follows that the length of the longer leg is √3 and the length of the hypotenuse is 2. 30

√3

1 =

Trigonometric Functions on a

Rectangular Coordinate System

x

y

q

Pick a point on the

terminal ray and drop a

perpendicular to the x-axis.

r y

x

The adjacent side is x

The opposite side is y

The hypotenuse is labeled r

This is called a

REFERENCE TRIANGLE. y

x

x

y

x

r

r

x

y

r

r

y

qq

qq

qq

cottan

seccos

cscsin

Trigonometric Ratios may be found by:

45 º

1

1

2Using ratios of special triangles

145tan

2

145cos

2

145sin

For angles other than 45º, 30º, 60º you will need to use a

calculator. (Set it in Degree Mode for now.)

We need a way to remember all of these ratios…