Mathematics Review A public service from: Runwaydata.org John R. Smith, MS, EdS.

Mathematics Review

A public service from:

Runwaydata.org

John R. Smith, MS, EdS

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Trigonometry Relationships

Please Obtain the Following:

• Pencil (or pen) & Scratch Paper

• Calculator - -TRIG FUNCTIONS (recommended)

• A Cup of Coffee (optional)

Please Draw this on your Scratch Paper:

Wasn’t that easy?

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Introduction - - UNIT CIRCLE

Why are we doing this?• Best Way - - TO STUDY

ANGLES• We Need - - ONLY TWO

QUANTITIES• All We Need - - 0 (for the

intersection of the x & y axes)

Please Draw Your Circle:

Start with a Name - - FOR THE “ORIGIN”

(0, 0)X-axis

Y-axis

(x, y)

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Unit Circle (Continued)

Unit Circle Conventions - -• Four 90° Quadrants - - 360°• Start From 0° - - QUADRANTS

ASSIGNED COUNTER-CLOCKWISE (I, II, III & IV)

• 90° Cardinal Point Boundaries - - SIGNIFICANT (Cosine, Sine & Tangent Definition (discussed later)]

Similar to a Compass - - 360° (around the circle)

0°

90°

180°

270°

III

III IV

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Best Way to Learn Angles - - THE CIRCLE

But why? Lets Explore:

Diameter - - ACROSS

Circumference - - AROUND (the distance around)

0° - - START HERE

360° - - END HERE

Radius

Learn these terms!

III

III IV

Know:• Circumference• Diameter• Radius

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The Concept of Pi - - A RATIO

Diameter - - ACROSS

Circumference - - AROUND (the distance around)

0° - - START HERE

360° - - END HERE

Radius

Π = pi = c/d = circumference/distance= 3.1459

Radius = ½ d

Important: For all circles (all diameters), the ratio of c to d is always 3.1459!

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For Example - - If Radius = 1, what is the Area? Solution Area = A = πr2, where 3.1459 * (1)2 = 3.1459

0° - - START HERE

360° - - END HERE

[r]adius

Π = pi = c/d = 3.1459

Radius (r) = ½ d = ½ * 2 = 1

Important: For all circles (all diameters), the ratio of c to d is always 3.1459!

Radius (r) = 1

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Another Example - - If DIAMETER = 2, what is the Area? Solution A = π (½ d)2 where [3.1459 * (½ * 2)2] = 3.1459

[d]iameter - - ACROSS 0° - - START HERE

360° - - END HERE

[r]adius

Π = pi = c/d = 3.1459

Radius (r) = ½ dImportant: For all circles (all diameters), the ratio of c to d is always 3.1459!

d =2

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The Right Triangle - - THE 30°/60°/90°!

x

yr

base

altitude

hypotenuse

r = 1

90°30°

60°

0

30°

0°

30°

30°

a

b

c

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The Right Triangle (revealed) - - THE 30°/60°/90°!

x

yr =1

base

altitude

Hypotenuse = 1

r = 1

90°30°

60°

30°

30°

30°

a

b

c = 1

θ = theta = 30°

y = ½ = .5 = SIN 30°

=== 866.2

3x COS 30°

θ

θ

θ

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Pythagorean Theorem

x

yr =1

base

altitude

Hypotenuse = 1

r = 1

90°30°

60°

30°

30°

30°

a

b

c = 1

θ = theta = 30°

y = ½ = .5 = SIN 30°

=== 866.2

3x COS 30°

θ

θ

θ

For any Given Angle - - HERE, 30° for example COS x2 + SIN y2 = 1Where, (.866)2 + (.5)2 = 1, orx2 + y2 = 1(.866)2 + (.5)2 = (1)2 = 1

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Angles, Radians & Pi (π)

r = 1

0

30°

0°θ

45°

60°90°

180°

270°

If π = 3.1459, then

Circumference/Diameter = 3.1459(for all circles). This means:c/d = π (always w/o exception)c = πd (here, 3.1459 * 2) = 6.283d = c/π (here, 6.283/3.1459) = 2

If the Diameter is 2, (radius must be 1);Circumference = Diameter * π = 2 * 3.1459 = 2π = 6.283 = 6.283 = 360° = 2π

diameter = 2

radius = 1

r =1

r =1

r =

1

r =1

r =

1

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Angles, Radians & Pi (π)

r = 1

0

30°

0°θ

45°

60°90°

180°

270°

diameter = 2

radius = 1

r =1

r =1

r =

1

r =1

r =

1

Because diameter is 2, (& radius =1);Circumference = Diameter * π = 2 * 3.1459 = 2π = 6.283 = 6.283 = 360° = 2π radians 360°/6.283 = 57°17’ = 1 radian

“fifty-seven degrees & 17 minutes”

If 360° = 2π Radians, then180° = π (&, 90° = π/2)270° = 3π/2 = (3 * π/2)

π/2

π

3π/2

2π

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Angles, Radians & Pi (π) [Continued]

618030

ππ=

°×°

r = 1

0

30°

0°θ

45°

60°90°

180°

270°

diameter = 2

radius = 1

r =1

r =1

r =

1

r =1

r =

1

418045

ππ=

°×°

318060

ππ=

°×°

ππ2

180360 =

°×°

} y = ½

866.02

3==x

5.02

1==y

θ 30°

1)866.0()5.0(

1cossin22

22

=+=

=+=

r

r θθ

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Angles, Radians &

Pi (π) [Continued]

618030

ππ=

°×°

r = 1

0

30°

0°θ

45°

60°90°

180°

270° = 3π/2

r =1

r =1 4180

45ππ

=°

×°

318060

ππ=

°×°

ππ2

180360 =

°×°

} y = ½

),)(sin,(cos yxyx

218090

ππ=

°×°

)707,.707)(.2

2,

2

2)(,)(sin,(cos yxyx

)866,.5)(.2

3,2

1)(,)(sin,(cos yxyx

)5.0,866)(.2

1,

2

3)(,)(sin,(cos yxyx

Pythagorean Theorem:x2 + y2 = 1Try it, it works! For example:(Cos 90°)2 + (Sin 90°)2 = 1(Cos 60°)2 + (Sin 60°)2 = 1(Cos 45°)2 + (Sin 45°)2 = 1 (Cos 30°)2 + (Sin 30°)2 = 1(Cos 0°)2 + (Sin 0°)2 = 1

x = 0.866

),)(sin,(cos yxyx

(1, 0)

(0, 1)

(-1, 0)

(0, -1)2

3

180270

ππ=

°×°

For example, cos 3π/2 = 0, &sin 3π/2 = -1

For example, cos π/2 = 0, & sin π/2 = 1, &Cosine 90° = 0, & Sine 90° = 1

(cos, sin)

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Unit Circle - - SIMPLIFIED

(1, 0)

(0, 1)

(0, -1)

(-1, 0)

III

III IV

Memory Aid - - CAST

(+, +)

(+, -)

C

AS

T

(-, -)

(-, +)

Cosines: Positive (sin & tan negative All signs: Positive (sin, cos & tan)Sines: Positive (cos & tan negative)Tangents: Positive (cos & sin negative)

Quiz:Cosine 0° = ?Sine 90° = ?Tangent 180° = ?Cos2 270° + Sin2 270° = ?Cos2 360° + Sin2 360° = ?Answers: Next slide!

x

yr

(x, y)(cos, sin)

Sin = y/rCos = x/rTangent = y/x

0°

90°

180°

270°

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Unit Circle - - SIMPLIFIED

(1, 0)

(0, 1)

(0, -1)

(-1, 0)

III

III IV

Memory Aid - - CAST

(+, +)

(+, -)

C

AS

T

(-, -)

(-, +)

Cosines: Positive (sin & tan negative All signs: Positive (sin, cos & tan)Sines: Positive (cos & tan negative)Tangents: Positive (cos & sin negative)

Quiz:Cosine 0° = 1Sine 90° = 1 Tangent 180° = y/x = 0/-1 = 0Cos2 270° + Sin2 270° = (0)2 + (-1)2 = 1Cos2 360° + Sin2 360° = (1)2 + (0)2 = 1

x

yr

(x, y)(cos, sin)

Sin = y/rCos = x/rTangent = y/x

0°

90°

180°

270°

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The Unit Circle - - A POWERFUL TOOL[Trigonometric Functions]: DRAW THIS!

Let’s Review - - PRACTICE!

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The Unit Circle - - A POWERFUL TOOL[Trigonometric Functions]: Assign Cardinal Point Coordinates!

III

III IV

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

0°180°

270°

90°

State as, “Cos 270° = 0& Sin 270° = -1”

(x, y) corresponds to (Cos x, Sin y)

State as, “Cos 0° = 1& Sin 0° = 0”

State as, “Cos 90° = 0& Sin 90° = 1”

State as, “Cos 180° = -1& Sin 0° = 0”

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The Unit Circle - - A POWERFUL TOOL[Trigonometric Functions]: Cosine & Sine - - INTUITIVE!

III

III IV

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

0°180°

270°

90°

Cosine 0° = 1Sine 0° = 0

Cosine 90° = 0Sine 90° = 1

Cosine 180° = -1Sine 180° = 0

Cosine 270° = 0Sine 270° = -1

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The Unit Circle - - A POWERFUL TOOL[Trigonometric Functions]: Memory Aid, “CAST”!

III

III IV

C

AS

T

All Signs - - POS +(+,+)

(cos, sin)

Cosines Only - - POS +(+,-)

(cos, sin)

Tangents Only - - POS +(-,-)

(cos, sin)

Sines Only - - POS +(-,+)

(cos, sin)

x

y[r]adius

θ

θ : “theta”Adjacent θ

opposite

θ

Hypotenuse

x

y

r

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The Unit Circle - - A POWERFUL TOOL[Trigonometric Functions]: Memorize!

III

III IV

x

y[r]adius

θ

Adjacent θ

opposite

θ

Hypotenuse x

y

r

θ

Sin = y/rCosine = x/rTangent = y/x

x2 + y2 = 1 (cos, sin)

30°

0°

45°

60°90°

)2

1,

2

3(

)2

2,

2

2(

Measurement - - DEGREES

)2

3,2

1(

)0,1(

1)2

3()

2

1( 22 =+

1)2

2()

2

2( 22 =+

)1,0(

1)2

1()

2

3( 22 =+

Cos 0° = 1 Sin 0° = 0

1)0()1( 22 =+

1)1()0( 22 =+

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The Unit Circle - - A POWERFUL TOOL[Trigonometric Functions]: Quiz!

III

III IV

x

y

r

θ

x2 + y2 = 1 (cos, sin)

30°

0°

45°

60°90°

)2

1,

2

3(

)2

2,

2

2(

Measurement - - DEGREES

)2

3,2

1(

)0,1(

1)2

3()

2

1( 22 =+

1)2

2()

2

2( 22 =+

)1,0(

1)2

1()

2

3( 22 =+

Cos 0° = 1 Sin 0° = 0

1)0()1( 22 =+

1)1()0( 22 =+

1. Cosine 30°?2. Sine 45°?3. (Cos)2 60° + (Sin)2 60° = ?4. Coordinates of the Origin?5. Quadrant Location 225°?

Answers: Next Slide!

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The Unit Circle - - A POWERFUL TOOL[Trigonometric Functions]: Quiz!

III

III IV

x

y

r

θ

x2 + y2 = 1 (cos, sin)

30°

0°

45°

60°90°

)2

1,

2

3(

)2

2,

2

2(

Measurement - - DEGREES

)2

3,2

1(

)0,1(

1)2

3()

2

1( 22 =+

1)2

2()

2

2( 22 =+

1)2

1()

2

3( 22 =+

Cos 0° = 1 Sin 0° = 0

1)0()1( 22 =+

1)1()0( 22 =+

1. Cosine 30°?2. Sine 45°?3. (Cos)2 60° + (Sin)2 60° = 14. Coordinates of the Origin? (0,0)5. Quadrant Location 225°? III

2

3

2

2(0, 1)

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The Unit Circle - - A POWERFUL TOOL[Trigonometric Functions]: Memorize!

III

III IV

x

y

r

θ

x2 + y2 = 1 (cos, sin)

π/6

0

π/4

π/3

)2

1,

2

3(

)2

2,

2

2(

Measurement - - RADIANS

)2

3,2

1(

)0,1(

1)2

3()

2

1( 22 =+

1)2

2()

2

2( 22 =+

)1,0(

1)2

1()

2

3( 22 =+

1)0()1( 22 =+

1)1()0( 22 =+

618030

ππ=

°×°

418045

ππ=

°×°

0180

0 =°

×°π

318060

ππ=

°×°

218090

ππ=

°×°

π/2

Quiz:1. Cos π/2 = ?2. Sin π/3 = ?3. Cos2 π/4 + Sin2 π/4 = ?4. Tan π/6? 5. Convert 360° to Radians

Bonus question:1. Convert 7π/6 to

degrees. Which quadrant?

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The Unit Circle - - A POWERFUL TOOL[Trigonometric Functions]: Memorize!

III

III IV

x

y

r

θ

x2 + y2 = 1 (cos, sin)

π/6

0

π/4

π/3

)2

1,

2

3(

)2

2,

2

2(

Measurement - - RADIANS

)2

3,2

1(

)0,1(

1)2

3()

2

1( 22 =+

1)2

2()

2

2( 22 =+

)1,0(

1)2

1()

2

3( 22 =+

1)0()1( 22 =+

1)1()0( 22 =+

618030

ππ=

°×°

418045

ππ=

°×°

0180

0 =°

×°π

318060

ππ=

°×°

218090

ππ=

°×°

π/2

Quiz Answers:1. Cos π/2 = 02. Sin π/3 = 3. Cos2 π/4 + Sin2 π/4 = 14. Tan π/6 = y/x = ½ / = .57745. Convert 360° to Radians:

2

3

2

3

ππ2

180360 =

°×°

Bonus question:1. Convert 7π/6 to

degrees. Which quadrant? III

oo

210180

6

7=×

ππ

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Mathematics Review• Congratulations, you have completed the mathematics review!• This PowerPoint presentation is in support of:

– Runwaydata.com ©• Aircraft Performance Explanation - - ISBN 978-3-8364-8343-8

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licensing)– Runwaydata.net ©

• Human Performance & Limitations (HPL)– 2009: Last weekend of APR, JUN, AUG & OCT

• Air Law & ATC Procedures– 2009: Last weekend of May, JUL, SEP, NOV

• The Above Schedule - - TO ACCOMMODATE ORLANDO TEST CENTER (first calendar M-F week of the new month).

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In the near future,….

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