Unit Plan Power Point

Unit Plan Power Point

Pythagorean TheoremSpecial right trianglesIntro to Trigonometry

Angles of Elevation and DepressionVectors

The Pythagorean Theorem

• A2+B2=C2

• The sum of the squares of the two shorter sides of a right triangle equals the square of the hypotenuse

Some very useful right triangles

• 45/45/90 – 1/1/√2• 30/60/90 – 1/√3/2 next section topic• x/x/x - 1/2/√5 …• Pythagorean “Triples”– 3/4/5, 5/12/13, 7/24/25, 9/40/41 (odd # sequence)– Multiples of these…

• To the whiteboard!

– 8/15/17, 20/21/29, 33/56/65, etc.– Multiples of these…

• To the whiteboard!

Worksheet 1 – Pythagorean Theorem problems

Find the value of x

1.

3. 4.

5. 6.

2.26 26x

483

3

2

x

x

x

x

x

40

96

14

7

258

6

Using the Pythagorean Theorem to see if a triangle is a right triangle, an acute triangle or an obtuse triangle.

If A2+B2>C2 then the triangle is acute

If A2+B2<C2 then the triangle is obtuse

Let’s look at a couple triangles to see why this is easy to visualize…

A A

B

C

C

Bobtuse

acute

Acute or Obtuse?

4/5/8? 4/5/6? 5/6/8?

1/2/2.5 1/3/3.5 2/3/4

7/8/10 12/13/18 8/9/12

Examples from the students…must follow the rules to make a triangle!!

Worksheet 2 – Independent Desk Work Name ___________

Solve for x – missing side (5)

Acute, obtuse, right triangle? (6)

SOL/SAT test prep (4)

For the following right triangles A/B/CSolve for x

1. 4/4/x x=____

2. 15/20/x x=____

3. 15/36/x x=____

4. 4/4√3/x x=____

5. 3/x/3√5 x=____

1. 4/5/6? ____ 2. 3/4/6? ____ 3. 1/2/√3 ____

4. 2/√5/3 ____ 5. 9/39/40 ____ 6. 5/13/14 ____

1. The longer leg of a 30/60/90 triangle is 6. What is the length of the hypotenuse? ____

2. The hypotenuse of a 45/45/90 is 9. What are the lengths of the two sides? ____ , ____

3. Which triangle is not a right triangle?27/36/45 5/5/5√2.6/.8/1.0 4/7/8

4. Complete the sequence3, 4, 5, 5, 12, 13, 7, 24, 25, __ , __ , __ , __ , __ , __ , __ , __ , __

Special Triangles

45/45/90 – 1/1/√2 30/60/90 – 1/√3/2

1 1

√2

1

12+12=(√2)2 1+1=2

2

√3

12+(√3)2=22 1+3=4

45°

45° 30°

60°

So what? You need to recognize these by looking for pairs of these numbers

What are the pairs?

Any time you see the same number twice that is a 45/45/90Any time you see a 1 & a 2 that is the 30/60/90Any time you see a √2 that’s usually a 45/45/90Any time you see a √3 that’s usually a 30/60/90Any time you see a multiple of any of these …

Recognize?

4/4/x 2/2√3/x 3/6/x 3/x/6

Worksheet 1 – Let’s do some problems together

x/6/6√2 x/4√3/8 2/2/x 5/5/x

10/x/20 5/x/10 12/12√3/x 7/x/14

45/45/90 with short side of 11

30/60/90 with short side of 11

45/45/90 with hypotenuse ~141

30/60/90 with long leg 1732

Combination triangles – Recognizing the Special Triangle Within

Cut an equilateral triangle into two with a bisector? 30/60/90sCut a 30/60/90 with an altitude? You get two smaller 30/60/90sCut a 45/45/90 with an altitude? You get two smaller 45/45/90s

Let’s look at all these on the whiteboard to see how the math works out

Worksheet 2 – Special triangles - You are on your own

1. 2.

3. 4.

5. 6.

60°45° 45°

45°

45°

45°

45°60°

45°

a ___b ___c ___

40

c

ba 36√2

a

b

c

c

a ___b ___c ___

60°

100

a ba ___b ___c ___

a ___b ___c ___

a ___b ___c ___

a ___b ___c ___

60°

60°

60°a

b

c

10

All angles 60° All heptagon sides = 8

a

b√6

a b

c

√6

Introduction to Trigonometry• Ratios of sides in similar triangles does not change. This is the basic idea behind trig.• The trigonometry we will be doing will be with right triangles.• There are three fundamental ratios in right triangle trigonometry –

Sine, Cosine, Tangent• Sine = Opposite side / Hypotenuse = O/H• Cosine = Adjacent side / Hypotenuse = A/H• Tangent = Opposite side / Adjacent side = O/A• The Law of Sines relates all the angles in all triangles

• The Law of Cosines looks a lot like the Pythagorean Theorem

• The Sine and Cosine are equal for any two complementary angles, i.e. Sin(60°) = Cos(30°); Sin(50°) = Cos (40°)This known at the Law of Sine and Cosine of Complementary Angles – this is the one you need to know for the SOL!

• Angles are expressed in both degrees and radians for trigonometry• All the way around a circle is 2π (360°), half circle is π (180°), π/2 = 90°

The “Unit Circle”

So let’s look at an example triangle and determine Sine, Cosine and Tangent

2

4√20=2√5

Sin (A) = 2/2√5 = 1/ √5 ≈ .4472 Cos (A) = 4/2√5 = 2/ √5 ≈ .8944 Tan (A) = 2/4 = 1/2 = .5

Sin (B) = 4/2√5 = 2/ √5 ≈ .8944 Cos (B) = 2/2√5 = 1/ √5 ≈ .4472 Tan (B) = 4/2 = 2.0B

A

One more with a familiar “triple”

3

45

Sin (A) = 3/5 = .6 Cos (A) = 4/5 = .8 Tan (A) = 3/4 = .75

Sin (B) = 4/5 = .8 Cos (B) = 3/5 = .6 Tan (B) = 4/3 = 1.3B

A

Law of Sines and Cosines of complementary angles.The law says that..

for two complementary angles the sine of one equals the cosine of the other

Let’s do some simple problems and get used to using your calculator

Degrees Radians Solve for the angleSin (30) = Sin (π/6) = Sin (x) = .5

Cos (30) = Sin (π/4) = Cos (x) = .6

Sin (45) = Sin (π/8) = Sin (x) = .7

Tan (45) = Cos (π/6) = Tan (x) = 1

Sin (60) = Cos (π/4) = Tan (x) = 2

Cos (60) = Cos (π/8) = Tan (x) = 3

Tan (60) = Tan (π/6) = Tan (x) = 4

Tan (0) = Tan (π/4) = Tan (x) = 99

Tan (90) = Tan (π/8) = Tan (x) = .01

Worksheet - Trigonometry – page 1 1. Label the all the sides

of the triangle from the information given

A

Sin (A) = .6

3. Label the all the sides of the triangle from the information given

A

Tan (A) = 1

2. Label the all the sides of the triangle from the information given

A

Cos (A) = .5

4. Label the all the sides of the triangle from the information given

A

Sin (A) = .2

Worksheet - Trigonometry – page 2

1. π/9 in radians = ____ in degrees2. π/3 in radians = ____ in degrees3. π/36 in radians = ____ in degrees4. π/4 in radians = ____ in degrees

5. Sin (π/9) = _____6. Cos (π/18) = _____7. Tan (π/18) = _____8. Sin (π/4) = _____

9. If Tan (x) = 1 what is Cos (x) =? ___10.If Tan (x) = .5 what is Sin (x) =? ___11.If Sin (x) = .6 what is Cos (x) =? ___12.If Cos (x) = (1/3) what is Tan (x) =? ___

(Express answer in 12 as a fraction)

13.

14.

15.

16.

What is the value of x?

x

x

x

18

x

70°

18

13

13

12

√245°

Angles of Elevation and Depression

Angle of Elevation: the angle up from you the viewer to an object higher than your eye line – the horizontal line straight ahead from youAngle of Depression: the angle down from you the viewer to an object lower than your eye line – the horizontal line straight ahead from you

Horizontal line away from your eyes – “eye line”

Angle of Elevation

Angle of Depression

How do you work these problems?

1. Impose (draw) a right triangle that represents the problem you are trying to solve

2. Label the triangle with the information you have been given in the problem

3. Use the trigonometry you have learned to figure out the answer.• Can you solve it with a “triple” quickly? Are there two sides given?

Do you recognize the two from a known “triple?”• Can you solve it with a common triangle quickly? Are there

recognizable ratios of the sides? Is there a telltale square root?• Can you solve it with the Pythagorean Theorem? Are there two

sides given?• Can you solve it with trigonometric ratios? Is there an angle and a

distance given? Is the angle a common angle for which you know the trigonometric ratios? (.5, .6, .75, .8 for Sin and Cos; .5, 1, 2 for Tan? Is the side given an oddball number you don’t remember seeing in a triple or in a special triangle?

Let’s do some real world problems…

1. How tall is the tree? You are 50’ away and are looking up at a 50° angle at the top of the tree.

2. You are standing on the tee box at the top of a cliff looking down at the green. The distance from the tee to the hole is 200 yards. The sign on the tee box says the hole is 141 yards straight ahead distance. What is the angle of depression?

3. You are in an airplane at 30,000’ and see a city off in the distance at a 60° angle of depression. How far away is the city from your position? From a spot directly below you on the ground?

4. You are a mile away from half-dome at Yosemite National Park at a position level to the base of the climbing face. You see a climber at an angle of elevation of 30°. How much of a climb does she have left? (Half dome is 4,373’ from the valley floor to the top)

Worksheet 2 – Independent Desk Work – Angles of Elevation & Depression – page 1Draw a triangle that represents the problem and then solve.Indicate what technique you used to solve the problem.

1. You are 55’ from a wind farm turbine windmill. Your angle of elevation to the hub of the windmill is 56.5°. Your eye level is 5.5’ above the ground. How far above the ground is the hub of the windmill?

2. You are on a ship at sea and see an airplane at a 45° angle of elevation. The ship’s radar says the airplane is 21.2 miles away along the horizon line (your eye line). To the nearest mile, how far away is the airplane from your ship?

3. You are standing on the top of Camelback Mountain near the city of Phoenix, Arizona. The top of the mountain is supposedly 1,200’ above the city. What is the angle of depression to downtown Phoenix that shows up as 8 miles away on the map?

Worksheet 2 – Independent Desk Work – Angles of Elevation & Depression – page 2

Work the following problems from the book. Show your drawing with answer squared in and explain your technique for solving the problem in 8 words or less.

#19 – Weather balloon problem

#22 – Statue of Liberty problem

#23 – Tallest flagpole in the world problem

#24 – Two buildings next to each other problem

#35 – Greatest elevation of the sun for Richmond problem

#19

#22

#23

#24

#35

Mr. Bechter Teaches the Vector

A vector has magnitude and direction

Magnitude is reflected in the length of the arrow

Direction is reflected by orientation of the arrow

+4 -9+2

-5

Vectors can be added and subtracted using trigonometry and the coordinate plane

Airplane speed/direction

Wind speed and direction

Resultant motion over ground of airplane

Speed and direction of swimmer in still water

Speed and direction of rip current at the beachRes

ultant m

otion over

ground of s

wimmer Baseball speed/direction w

ind

Resultant motion over ground of baseball

Lets put some numbers on these!

Airplane speed/direction

Wind speed and direction

Resultant motion over ground of airplane

Speed and direction of swimmer in still water

Speed and direction of rip current at the beachRes

ultant m

otion over

ground of s

wimmer Baseball speed/direction w

ind

Resultant motion over ground of baseball

550 mph

100 mph

Speed over ground?

90 mph

10 mph

Speed over ground?

3 mph

8 mph

Resulta

nt speed an

d

directi

on over g

round?

Let’s do a few from the book using the coordinate plane…

Chapter Review8-1: Pythagorean Theorem

• A2+B2=C2

• Odd # triples• Other triples

8-2: Special Right Triangles• 30/60/90• 45/45/90• 1/√3/2• 1/1/ √2• 1/2/ √5

8-3: Trigonometry• Definitions of Sin, Cos, Tan• Expressing Sin, Cos, Tan in terms of ratios for a given triangle• Law of Sin e& Cosine of Complementary Angles• Using your calculator to solve for angle• Using your calculator to solve for length of side

8-4: Angles of Elevation & Depression• Use all approaches available to solve real world problems

8-5: Vectors• Use all approaches available to solve real world problems

Suggested textbook problems to study in preparation for the test…

• All of the Chapter Test questions (1-24)

• All the multiple choice questions in the Cumulative Test Prep (1-10)

• #13, #16, #19, #21 in the Cumulative Test Prep

Suggested other study problems

• All of the independent worksheet problems from the last 5 classes

• Your notes on proofs of the Pythagorean Theorem from the Khan Academy videos