MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions

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MTH 112Elementary Functions Chapter 5The
Trigonometric Functions

• Section 3 Trigonometric Functions of Any Angle

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Two Views of an Angle

• Geometric
• The union of two rays with a common endpoint.
• Measurement is from 0? to 180? (limited range)

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Two Views of an Angle

• Geometric
• A Rotation
• Position a ray on the positive x-axis with the
  endpoint at the origin. This is the initial side
  of the angle.
• Rotate a copy of this ray around the origin to
  produce the terminal side of the angle.
• Measurement specifies the amount direction of
  rotation.

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Signed Angle Measurement
(Rotational View of an Angle)

• Positive Angles
• Counterclockwise rotation.

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Signed Angle Measurement
(Rotational View of an Angle)

• Positive Angles
• Negative Angles
• Clockwise rotation.

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Signed Angle Measurement
(Rotational View of an Angle)

• Positive Angles
• Negative Angles
• Right Angles
• 90?
• -90?
• 270?
• -270?
• 450?
• etc.

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Signed Angle Measurement
(Rotational View of an Angle)

• Positive Angles
• Negative Angles
• Right Angles
• Rays
• 0?
• 360?
• -360?
• etc.

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Signed Angle Measurement
(Rotational View of an Angle)

• Positive Angles
• Negative Angles
• Right Angles
• Rays
• Lines
• 180?
• -180?
• 540?
• etc.

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Signed Angle Measurement
(Rotational View of an Angle)

• How many measurements can a single angle
  represent?
• ?
• How are these measurements related?
• n 360k?
• 0? n lt 360?
• k ? non-negative integers

Coterminal Angles with the same terminal side.
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Trigonometric Functions of Acute Angles of
Rotation

• sin ? y / r
• cos ? x / r
• tan ? y / x
• sec ? r / x
• csc ? r / y
• cot ? x / y

r
x2 y2 r2
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Trigonometric Functions of Any Angle
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Trigonometric Functions of Any Angle First
Quadrant

• sin ? y / r
• cos ? x / r
• tan ? y / x
• sec ? r / x
• csc ? r / y
• cot ? x / y

x gt 0 y gt 0 ? All six functions are positive.
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Trigonometric Functions of Any Angle Second
Quadrant

• sin ? y / r
• cos ? x / r
• tan ? y / x
• sec ? r / x
• csc ? r / y
• cot ? x / y

(x, y)
r
y
x
x2 y2 r2
x lt 0 y gt 0 ? cos ? is negative sin ? is
positive.
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Trigonometric Functions of Any Angle Third
Quadrant

• sin ? y / r
• cos ? x / r
• tan ? y / x
• sec ? r / x
• csc ? r / y
• cot ? x / y

?
x
y
r
(x, y)
x2 y2 r2
x lt 0 y lt 0 ? cos ? sin ? negative.
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Trigonometric Functions of Any Angle Fourth
Quadrant

• sin ? y / r
• cos ? x / r
• tan ? y / x
• sec ? r / x
• csc ? r / y
• cot ? x / y

?
x
y
r
(x, y)
x2 y2 r2
x gt 0 y lt 0 ? cos ? is positive sin ? is
negative.
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Summary of the Signs of the Trig Functions by
Quadrant
sin gt 0 cos lt 0 tan lt 0
All positive
sin lt 0 cos lt 0 tan gt 0
sin lt 0 cos gt 0 tan lt 0
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Reference Angle
(-x, y)

• The first quadrant angle containing the point
• ( x, y )
• Where (x, y) is on the terminal side of the
  angle.

(x, y)
(x, y)
?
?
(-x, -y)
(x, -y)
?
?
(x, y)
(x, y)
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Using Reference Angles to Find Trig Values of
Multiples of 30, 45, and 60 Angles.
The same idea can be applied to the other two
quadrants.
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Given one trigonometric value of an angle and the
quadrant of the terminal side of the angle, find
the other five trigonometric values.

• Write the given value as a fraction.
• - What about negative signs?
• Draw a right triangle, specify the coterminal
  angle (?), and use the fraction to determine the
  lengths of two sides relative to the indicated
  angle.
• Determine the other five trigonometric values
  using the definitions.
• Determine the signs of the other five
  trigonometric values using the indicated quadrant.

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Given one trigonometric value of an angle and the
quadrant of the terminal side of the angle, find
the other five trigonometric values example!
Problem If sec ? -2.5 and ? is in the second
quadrant, find sin ?, cos ?, tan ?.
2.5 5/2
5
2
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Trig Values of Angles with the Terminal Side on
an Axis
Case 1 Positive y-Axis 90 360k (k is any
non-negative integer)
sin 90 r / r 1 cos 90 0 / r 0 tan 90
r / 0 .. undefined !
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Trig Values of Angles with the Terminal Side on
an Axis
Case 2 Negative x-Axis 180 360k (k is
any non-negative integer)
sin 180 0 / r 0 cos 180 -r / r -1 tan
180 0 / -r 0
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Trig Values of Angles with the Terminal Side on
an Axis
Case 3 Negative y-Axis 270 360k (k is
any non-negative integer)
sin 270 -r / r -1 cos 270 0 / r 0 tan
270 -r / 0 .. undefined !
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Trig Values of Angles with the Terminal Side on
an Axis
Case 4 Positive x-Axis 0 360k (k is any
non-negative integer)
sin 0 0 / r 0 cos 0 r / r 1 tan 0 0
/ r 0
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Trig Values of Angles with the Terminal Side on
an Axis
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Review of Known Acute Angles