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# Welcome to Trigonometry! - PowerPoint PPT Presentation

Welcome to Trigonometry!. We’ll be “Getting’ Triggy” with these concepts… 6.1: find exact values of trigonometric functions (5-1) 6.2: find coterminal and reference angles and to covert between units of angle measure (5-1)

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Welcome to Trigonometry!

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### Welcome to Trigonometry!

We’ll be “Getting’ Triggy” with these concepts…

• 6.1: find exact values of trigonometric functions (5-1)

• 6.2:find coterminal and reference angles and to covert between units of angle measure (5-1)

• 6.3: solve for missing values in right triangles (5-4, 5-5)

• 6.4:use the law of sines and cosines and corresponding area formulas (5-6)

• 6.5:use the ambiguous case of the law of sines to solve problems (5-7)

### 6.2 find coterminal and reference angles and to convert between units of angle measure (5-1)

In this section we will answer…

• How are angles measured in Trig?

• What are the different units of angle measure within degree measurement?

• What does it mean for angles to be co-terminal?

• How can I find a reference angle?

### Angles and Their Measures

• From Geometry:

### In Trig

• Angles are always placed on the coordinate plane.

• The vertex is at the origin and one side (theinitial side) lies along the x-axis.

• The other side (the terminal side) lies in a quadrant or on another axii.

• This is called Standard Position.

### Angle Direction:

• Angles can be measured in two directions.

• Counter-clockwise is positive.

• Clockwise is negative.

### Degree Measurement:

• One full rotation = _________________.

• The circle has been cut into 360 equal pieces.

• Measure of less than a degree can be shown 2 ways:

• Decimal pieces: 55.75º

• Minutes and seconds: used for maps 103º 45’ 5”

• Each degree is divided into 60 minutes.

• Each minute is divided into 60 seconds.

• 1º = 60’ = 3600”

### Change 183.47

P280 #19 – 65 odd

### Translating Rotations to Degrees

• Give the angle measure which is represented by each rotation:

• 5.5 rotations clockwise

• 3.3 rotations counterclockwise

### Coterminal Angles

• Angles in standard position which share the same terminal side.

150º

- 210º

### Finding Coterminal Angles

• Simply add or subtract 360º as many times as you like.

• To write a statement to find EVERY angle coterminal with a certain angle:

• Identify all the angles which are coterminal with the given angle. Then find one positive and one negative coterminal angle.

• 86º

• 294º

• If each angle is in standard position, a) State the quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0º and 360º.

• 595º

• -777º

### Reference Angle:

• The acute angle formed by the terminal side of an angle in standard position and the x-axis.

• The quickest route to the x-axis.

### Recap:

• How are angles measured in Trig?

• What are the different units of angle measure within degree measurement?

• What does it mean for angles to be co-terminal?

• How can I find a reference angle?

### Homework:

• P280 #19 – 65 odd

• Portfoliodue Thursday 4/14

### 6.3: solve for missing values in right triangles (5-4, 5-5)

In these sections we will answer…

How are the 6 trig ratios expressed in geometry? In trig?

How can I use these relationships to solve triangle problems?

### Right Triangles in Geometry:

∆ ABC

Used the 3 basic trig ratios:

sin A, cos A and tan A.

SOH-CAH-TOA

Now we will add 3 reciprocal ratios:

csc A, sec A and cot A.

p 288

### Right Triangles in Trig:

Angles are in Standard Position in the Unit Circle.

1

1

-1

-1

### Try Some…

The terminal side of angle θ in standard position contains (8,-15), find the 6 trig ratios.

Now find the angle.

### If the csc θ = -2 and θ lies in QIII, find all 6 trig values.

Now find the angle.

### If the tan θ = -2 and θ lies in QII, find all 6 trig values.

Now find the angle.

### Homework:

P 288 #11 – 25 odd

P 296 #15 – 45 odd and 49

### 6.1: find exact values of trigonometric functions (5-2/5-3)

In this standard we will…

• Review the side relationships of 30°-60°-90° and 45°-45°-90° triangles.

• Build trig ratios based 30°-60°-90° and 45°-45°-90° triangles.

### Special Triangles from Geometry:

Build a chart to show all 6 trig ratios.

### 6.3 solve for missing values in right triangles (5-4)

In this standard we will answer…

• How can right triangle relationships by used to solve problems?

• If A = 37º and b = 6, solve the rest of the triangle.

### The apothem of a regular pentagon is 10.8 cm. Answer the following.

• Find the radius of the circumscribed circle.

• What is the length of one side of the pentagon?

• Find the perimeter of the pentagon.

r

a = 10.8 cm

### p 303

• Mr. Fleming is flying a kite. Ms Case notices the string makes a 70˚ angle with the ground. “I know the string is 65 meters long,” says Ms Case. “I wonder how far is the kite above the ground?”

• Ranger Gladd sights a fire from his fire tower in Alvarez National forest. He finds an angle of depression to the fire of 22˚. If the tower is 75 meters tall, how far is the fire from the base of the tower?

### Partner Solve:

• ONE piece of paper.

• One person solves then second person checks and either praises or coaches.

• Change jobs.

• Do p 301 #1 – 9 all

### Homework:

• LEARN YOUR SPECIAL TRIANGLES or UNIT CIRCLE!

• P 303 #11-29 odd

### 6.3: solve for missing values in right triangles (5-5)

In this section we will answer…

What can I do to solve if I don’t know any angles, just sides?

### Inverse/arc trig ratios:

To show you want to inverse or “undo” a trig ratio in order to get an angle there are two notations:

### Architecture:

Many cities place restrictions on the height and placement of skyscrapers in order to protect residents from completely shaded streets. If a 100-foot building casts an 88-foot shadow, what is the angle of elevation to the sun?

### Partner Workout!

P 309 #1 –14 all

One piece of paper, take turns solving.

If you aren’t solving you are the cheerleader/spotter. Encourage and save them from falling on their face.

### Homework:

• LEARN YOUR SPECIAL TRIANGLES or UNIT CIRCLE!

• P309 #15 – 45 odd

### Section 5-6: The Law of Sines

In this section we will answer…

Is there some way to solve triangles that aren’t right triangles?

How can I find the area of a triangle if I don’t know its height?

### The Law of Sines

Up to now we have worked with RIGHT triangles, but what about other kinds?

### The Law of Sines

Let’s look at ∆ABC

Then the following is true:

B

a

c

C

A

b

### Using the Law of Sines

How does it work?

How many values do you have to be provided with?

When won’t it work?

### From Geometry:

AAS: A = 40º, B = 60º, and a = 20

SAS: b = 10, C = 50º, and a = 14

ASA: c = 2.8, A = 53º, and B = 61º

One more for fun! b = 16, A = 42º, and c = 12

### Area of a Triangle

How did we find the area of triangles is geometry?

You can now find the area of ANY triangle whether or not the height is given!

B

a

c

C

A

b

h

Watch this!

B

a

c

C

A

b

h

### Finding the Area of a SSS Triangle:

Can we do this? How?

### Try a couple…

p 316 #20, 22, 24 and 26

### Homework:

p 316 #11 – 33 odd

Mini-Quiz! on special triangle values everyday!!!

### Section 5-7: The Ambiguous Case for the Law of Sines

In this section we will answer…

When can I use Law of Sines?

Is there ever a case where ASS actually WORKS?

How can I determine when I can use this really inappropriate acronym?

Do I have to memorize the chart?

### When can I use Law of Sines?

If I have…

AAS or ASA, always works!

If I have…

SAS or SSS, never works!

Exploration:

### Our Nifty, Triangle Info Sheet!

Do I have to memorize the chart?

### Try a few…

p 324 #11, 16, 19, 23 and 25

### Homework:

Mini-Quiz! on special triangle values tomorrow!

Unit Test! on Tuesday

1 = sin 2 = cos 3 = tan 4 = csc 5 = sec 6 = cot

1 = 0º 11 = 240º21 = -150º

2 = 30º12 = 270º22 = -180º

3 = 45º13 = 300º23 = -210º

4 = 60º14 = 330º24 = -270º

5 = 90º15 = 360º25 = -300º

6 = 120º16 = -30º 26 = -330º

7 = 135º17 = -45º27 = -360º

8 = 150º18 = -60º28 = 225º

9 = 180º19 = -90º29 = 315º

10 = 210 º20 = -120º30 = -225º

### Section 5-8: The Law of Cosines

In this section we will answer…

And then there is AAA, is that good for anything?

How can this be used for something real?

### Let’s do some…

A = 40º, b = 3 and c = 2

### Another…

a = 8, b = 9, c = 7

### Okay, let’s mix it up!

Solve for the missing values in each triangle.

a = 38, b = 25 and C = 90º

A = 75º, B = 50º and a = 7

A = 145º, a = 5, b = 10

A = 56.7º, B = 33.3º and c = 45.5

C = 55º, b = 5.6 and c = 5.9

None

### Finding the Area of a SSS Triangle:

Can we do this? How?

202 ft

B

C

82.5º

124.5º

180.25 ft

201.5 ft

75º

D

161º

97º

158 ft

A

E

125 ft

202 ft

B

C

82.5º

124.5º

180.25 ft

201.5 ft

75º

D

161º

97º

158 ft

A

E

125 ft

### Homework:

P331 #11 – 29 odd

Unit 6 TEST!!! Wednesday!.