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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)

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Welcome to trigonometry
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
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
Angles and Their Measures between units of angle measure (5-1)

  • From Geometry:


In trig
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
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
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
Change -16.75 between units of angle measure (5-1)


Change 183 47
Change 183.47 between units of angle measure (5-1)

P280 #19 – 65 odd


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


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


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


Translating rotations to degrees
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
Coterminal Angles between units of angle measure (5-1)

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

150º

- 210º


Finding coterminal angles
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
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
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
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
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
5-Minute Check Lesson 5-2B 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 5 5
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
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
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
*Be Careful with Reciprocal Ratios* quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0


Try some more
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
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
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 2 and lies in qiii find all 6 trig values
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 2 and lies in qii find all 6 trig values
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.


Homework1
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 up1
WARM-UP: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0


Homework2
Homework: quadrant in which the terminal side lies b) Determine a coterminal angle that is between 0


6 1 find exact values of trigonometric functions 5 2 5 3
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
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
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
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
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 62 and c 24 solve 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
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
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
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


Homework3
Homework Answer the following.:

  • LEARN YOUR SPECIAL TRIANGLES or UNIT CIRCLE!

  • P 303 #11-29 odd


Homework4
Homework: Answer the following.


6 3 solve for missing values in right triangles 5 5
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
Solve each equation if Answer the following.


Inverse arc trig ratios
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
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
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.


Homework5
Homework Answer the following.:

  • LEARN YOUR SPECIAL TRIANGLES or UNIT CIRCLE!

  • P309 #15 – 45 odd


Section 5 6 the law of sines
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
The Law of Sines Answer the following.

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


The law of sines1
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
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
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
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 triangle1
Area of a Triangle Answer the following.

Watch this!

B

a

c

C

A

b

h


Area of a triangle for sas
Area of a Triangle for SAS Answer the following.


What if i don t have 2 sides what if i have 2 angles let s say i know b c and a
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
Finding the Area of a SSS Triangle: Answer the following.

Can we do this? How?


For aas or asa
For AAS or ASA: Answer the following.


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


Try a couple
Try a couple… Answer the following.

p 316 #20, 22, 24 and 26


Homework6
Homework: Answer the following.

p 316 #11 – 33 odd

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


Section 5 7 the ambiguous case for the law of sines
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
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
Is there ever a case where ASS actually WORKS? Answer the following.

Exploration:


Our nifty triangle info sheet
Our Nifty, Triangle Info Sheet! Answer the following.

Do I have to memorize the chart?


Try a few
Try a few… Answer the following.

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


Homework7
Homework: Answer the following.

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 = 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 the law of cosines
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?



Let s do some
Let’s do some… cot

A = 40º, b = 3 and c = 2


Another
Another… cot

a = 8, b = 9, c = 7


Okay let s mix it up
Okay, let’s mix it up! cot

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
Answers: cot

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 triangle1
Finding the Area of a SSS Triangle: cot

Can we do this? How?


Our hero s formula it saves us1
Our cotHero’s Formula!It saves us!


Okay now something interesting find the area of this
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 46 471 6 sq 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


Homework9
Homework something interesting…:

P331 #11 – 29 odd

Unit 6 TEST!!! Wednesday!.


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