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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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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 between units of angle measure (5-1)

• From Geometry:

In Trig between units of angle measure (5-1)

• 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: between units of angle measure (5-1)

• Angles can be measured in two directions.

• Counter-clockwise is positive.

• Clockwise is negative.

Degree Measurement: between units of angle measure (5-1)

• 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 -16.75 between units of angle measure (5-1)

Change 183.47 between units of angle measure (5-1)

P280 #19 – 65 odd

Change 29 between units of angle measure (5-1)º 30’ 60”

Change 103 between units of angle measure (5-1)º 12’ 42”

Degrees on the Coordinate Plane: between units of angle measure (5-1)Unit Circle

Translating Rotations to Degrees between units of angle measure (5-1)

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

• 5.5 rotations clockwise

• 3.3 rotations counterclockwise

Coterminal Angles between units of angle measure (5-1)

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

150º

- 210º

Finding Coterminal Angles between units of angle measure (5-1)

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

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

Reference Angle: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

• 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: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

• 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: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

• P280 #19 – 65 odd

• Portfoliodue Thursday 4/14

5-Minute Check Lesson 5-2A quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

5-Minute Check Lesson 5-2B quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

6.3 quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0: 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: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

∆ 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.

Solving Using Right Triangles: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

*Be Careful with Reciprocal Ratios* quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

Try some more… quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

p 288

Right Triangles in Trig: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

Angles are in Standard Position in the Unit Circle.

1

1

-1

-1

Try Some… quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

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

Now find the angle.

If the csc quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0θ = -2 and θ lies in QIII, find all 6 trig values.

Now find the angle.

If the tan quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0θ = -2 and θ lies in QII, find all 6 trig values.

Now find the angle.

Homework: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

P 288 #11 – 25 odd

P 296 #15 – 45 odd and 49

WARM-UP: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

Homework: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

6.1 quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0: 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: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

Build a chart to show all 6 trig ratios.

Special Triangles on the Unit Circle: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

6.3 solve for missing values in right triangles (5-4) quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

In this standard we will answer…

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

Let’s start with some triangles… quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

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

If B = 62º and c = 24, solve the triangle. quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0

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 Answer the following.

• 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 Answer the following.:

• 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 Answer the following.:

• LEARN YOUR SPECIAL TRIANGLES or UNIT CIRCLE!

• P 303 #11-29 odd

Homework: Answer the following.

6.3 Answer the following.: 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?

Solve each equation if Answer the following.

Inverse/arc trig ratios: Answer the following.

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

Architecture: Answer the following.

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! Answer the following.

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 Answer the following.:

• LEARN YOUR SPECIAL TRIANGLES or UNIT CIRCLE!

• P309 #15 – 45 odd

Section 5-6 Answer the following.: 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 Answer the following.

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

The Law of Sines Answer the following.

Let’s look at ∆ABC

Then the following is true:

B

a

c

C

A

b

Using the Law of Sines Answer the following.

How does it work?

How many values do you have to be provided with?

When won’t it work?

From Geometry: Answer the following.

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 Answer the following.

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

Area of a Triangle Answer the following.

Watch this!

B

a

c

C

A

b

h

Area of a Triangle for SAS Answer the following.

What if I don’t have 2 sides? Answer the following.What if I have 2 angles? Let’s say I know b, C and A.

Finding the Area of a SSS Triangle: Answer the following.

Can we do this? How?

For AAS or ASA: Answer the following.

Our Answer the following.Hero’s Formula!It saves us!

Try a couple… Answer the following.

p 316 #20, 22, 24 and 26

Homework: Answer the following.

p 316 #11 – 33 odd

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

Section 5-7 Answer the following.: 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? Answer the following.

If I have…

AAS or ASA, always works!

If I have…

SAS or SSS, never works!

Is there ever a case where ASS actually WORKS? Answer the following.

Exploration:

Our Nifty, Triangle Info Sheet! Answer the following.

Do I have to memorize the chart?

Try a few… Answer the following.

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

Homework: Answer the following.

Mini-Quiz! on special triangle values tomorrow!

Unit Test! on Tuesday

1 = 0º 11 = 240º 21 = -150º

2 = 30º 12 = 270º 22 = -180º

3 = 45º 13 = 300º 23 = -210º

4 = 60º 14 = 330º 24 = -270º

5 = 90º 15 = 360º 25 = -300º

6 = 120º 16 = -30º 26 = -330º

7 = 135º 17 = -45º 27 = -360º

8 = 150º 18 = -60º 28 = 225º

9 = 180º 19 = -90º 29 = 315º

10 = 210 º 20 = -120º 30 = -225º

Section 5-8 cot: The Law of Cosines

In this section we will answer…

What about SAS?

How about SSS?

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

How can this be used for something real?

A = 40º, b = 3 and c = 2

Another… cot

a = 8, b = 9, c = 7

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

Answers: cot

A = 56.7º, B = 33.3º and c = 45.5

C = 55º, b = 5.6 and c = 5.9

None

Can we do this? How?

Our cotHero’s Formula!It saves us!

Okay, now something interesting…Find the area of THIS!

202 ft

B

C

82.5º

124.5º

180.25 ft

201.5 ft

75º

D

161º

97º

158 ft

A

E

125 ft

Answer: something interesting…46,471.6 sq 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 something interesting…:

P331 #11 – 29 odd

Unit 6 TEST!!! Wednesday!.