mathematics review a public service from: runwaydata.org john r. smith, ms, eds
TRANSCRIPT
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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