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)

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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)


Warm-up:


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 -16.75


Change 183.47

P280 #19 – 65 odd


Change 29º 30’ 60”


Change 103º 12’ 42”


Degrees on the Coordinate Plane: Unit Circle


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


5-Minute Check Lesson 5-2A


5-Minute Check Lesson 5-2B


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.


Solving Using Right Triangles:


*Be Careful with Reciprocal Ratios*


Try some more…

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


WARM-UP:


Homework:


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.


Special Triangles on the Unit Circle:


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?


Let’s start with some triangles…

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


If B = 62º and c = 24, solve 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


Homework:


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?


Solve each equation if


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:


Evaluate each expression assuming a Quadrant I angle.


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


Area of a Triangle

Watch this!

B

a

c

C

A

b

h


Area of a Triangle for SAS


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


Finding the Area of a SSS Triangle:

Can we do this? How?


For AAS or ASA:


Our Hero’s Formula!It saves us!


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!


Is there ever a case where ASS actually 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


Homework:


Section 5-8: 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?


Enter the conquering hero!


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


Answers:

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?


Our Hero’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: 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:

P331 #11 – 29 odd

Unit 6 TEST!!! Wednesday!.


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