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 (the initial 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

Portfolio due 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*

1sin csc

1csc

sin

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 oddP 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:

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

1 0 11 240 21 150

2 30 12 270 22 180

3 45 13 300 23 210

4 60 14 330 2

4 270

5 90 15 360 25 300

6 120 16 30 26 330

7 135 17 45 27 360

8 150 18 60 28 225

9 180 19 90 29 31

5

10 210 20 120 30 225

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.

a = 10.8 cmr

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

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

1 0 11 240 21 150

2 30 12 270 22 180

3 45 13 1020 23 210

4 60 14 330

24 270

5 630 15 720 25 300

6 120 16 30 26 1200

7 135 17 945 27 585

8 150 18 60 28 225

9 180 19 90 2

9 315

10 210 20 120 30 225

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 0 360x

tan 3

cos 0

tan 1

x

x

x

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:

1sin or arcsin

Evaluate each expression assuming a Quadrant I angle.

1

1

4cos(arccos )

5

2sec(cos )

5

5tan(cos )

13

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

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

1 0 11 240 21 150

2 30 12 270 22 180

3 45 13 1020 23 210

4 60 14 330

24 270

5 630 15 720 25 300

6 120 16 30 26 1200

7 135 17 945 27 585

8 150 18 60 28 225

9 180 19 90 2

9 315

10 210 20 120 30 225

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:

Ab

ca

C

B

sin sin sina b cA B C

Using the Law of Sines

How does it work?

How many values do you have to be provided with?

When won’t it work?

sin sin sina b cA B C

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!

Ab

ca

C

B

h

Area of a Triangle

Watch this!

Ab

ca

C

B

h

Area of a Triangle for SAS

1Area sin

21

sin21

sin2

K bc A

K ac B

K ab C

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

1Area sin

2K bc A

Finding the Area of a SSS Triangle:

3, 4 and 5a b c Can we do this? How?

For AAS or ASA:2

2

2

1 sin sin2 sin

1 sin sin2 sin

1 sin sin2 sin

c A BK

C

a B CK

A

b A CK

B

Our Hero’s Formula!It saves us!

where

known as the semi-perimeter

1( )2

( )( )( )

s a b c

K s s a s b s c

Find the area of a triangle whose sides are

30, 50 and 56.a b c

Try a couple…

p 316 #20, 22, 24 and 26

Homework:

p 316 #11 – 33 odd

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

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

1 0 11 240 21 150

2 30 12 270 22 180

3 45 13 1020 23 210

4 60 14 330 2

4 270

5 630 15 720 25 300

6 120 16 30 26 1200

7 135 17 945 27 585

8 150 18 60 28 225

9 180 19 90 29

315

10 210 20 120 30 225

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 = 240º 21 = -150º2 = 30º 12 = 270º 22 = -180º3 = 45º 13 = 300º 23 = -210º4 = 60º 14 = 330º 24 = -270º5 = 90º 15 = 360º 25 = -300º6 = 120º 16 = -30º 26 = -330º7 = 135º 17 = -45º 27 = -360º8 = 150º 18 = -60º 28 = 225º9 = 180º 19 = -90º 29 = 315º10 = 210 º 20 = -120º 30 = -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!

(except AAA which NEVER, EVER works!)

Works f or everything

2 2 2

2 2

The Law of Cosines

2 cos

a b c bc Ab a

2

2 2 2

2 cos 2 cos

c ac Bc a b ab C

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:

3, 4 and 5a b c Can we do this? How?

Our Hero’s Formula!It saves us!

1where ( )

2known as the semi-perimeter

( )( )( )

s a b c

K s s a s b s c

Find the area of a triangle whose sides are

30, 50 and 56.a b c

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

A

B C

75º

124.5º82.5º

201.5 ft

125 ft

180.25 ft

158 ft

202 ft

E

D

97º 161º

Answer: 46,471.6 sq ft

A

B C

75º

124.5º82.5º

201.5 ft

125 ft

180.25 ft

158 ft

202 ft

E

D

97º 161º

Homework:

P331 #11 – 29 odd

Unit 6 TEST!!! Wednesday!.