Trigonometric Functions of Any Angle 4'4

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Trigonometric Functions of Any Angle 4.4
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Definitions of Trigonometric Functions of Any
Angle

• Let ? is be any angle in standard position, and
  let P (x, y) be a point on the terminal side of
  ?. If r x2 y2 is the distance from (0, 0)
  to (x, y), the six trigonometric functions of ?
  are defined by the following ratios.

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Example
Let P (-3, -4) be a point on the terminal side
of ?. Find each of the six trigonometric
functions of ?.
Solution The situation is shown below. We need
values for x, y, and r to evaluate all six
trigonometric functions. We are given the values
of x and y. Because P (-3, -4) is a point on
the terminal side of ?, x -3 and y -4.
Furthermore,
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Example Cont.

• Solution
• Now that we know x, y, and r, we can find the six
  trigonometric functions of ?.

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Example
Let tan ? -2/3 and cos ? gt 0. Find each of the
six trigonometric functions of ?.
We have to be in Quadrant IV
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The Signs of the Trigonometric Functions
All Students Take Calculus
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Definition of a Reference Angle

• Let ? be a nonacute angle in standard position
  that lies in a quadrant. Its reference angle is
  the positive acute angle ? prime formed by the
  terminal side or ? and the x-axis.

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Example

• Find the reference angle ?, for the following
  angle ? 315º
• Solution
• ? 360º - 315º 45º

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Example
Find the reference angles for
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Using Reference Angles to Evaluate Trigonometric
Functions

• The values of a trigonometric functions of a
  given angle, ?, are the same as the values for
  the trigonometric functions of the reference
  angle, ?, except possibly for the sign. A
  function value of the acute angle, ?, is always
  positive. However, the same functions value for ?
  may be positive or negative.

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A Procedure for Using Reference Angles to
Evaluate Trigonometric Functions

• The value of a trigonometric function of any
  angle ? is found as follows
• Find the associated reference angle, ?, and the
  function value for ?.
• Use the quadrant in which ? lies to prefix the
  appropriate sign to the function value in step 1.

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Example
Use reference angles to find the exact value of
the following trigonometric functions.
a. sin 135
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Example cont.
Solution
The function value for the reference angle is sin
45º ?2 / 2. Step 2 Use the quadrant in
which è lies to prefix the appropriate sign to
the function value in step 1. The angle 135º lies
in quadrant II. Because the sine is positive in
quadrant II, we put a sign before the function
value of the reference angle. Thus, sin135?
sin45??2 / 2
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Example

• Evaluate