Trigonometric Functions of Any Angle

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Section 4.4

• Trigonometric Functions of Any Angle

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Overview

• In this section we will find trigonometric values
  for any angle.
• Doing so requires to consider rotations for
  circles centered at the origin but not with a
  radius of 1.
• We will use right triangle trigonometry to write
  the appropriate ratios.

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Definitions
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Examples

1. The point (-9, 12) is on the terminal side of an
   angle ?. Find the exact value of each of the six
   trigonometric functions of ?.
2. The point (-5, -4) is on the terminal side of an
   angle ?. Find the exact value of each of the six
   trigonometric functions of ?.

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Trig Values And Their Signs

• Recall the signs of the coordinates of x and y
  and each of the four quadrants
• The six trigonometric values follow the same
  rules and patterns.

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Students All Take Calculus
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Examples

1. Given that cos ? -4/5 and ? is in Quadrant II,
   find the remaining five trigonometric values of
   ?.
2. Given that tan ? 12/5 and cos ? lt 0, find the
   remaining five trigonometric values of ?.

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Reference Angles

• A reference angle is a positive acute angle
  formed by the terminal side and the x-axis.
• To find a reference angle, first find find the
  coterminal angle for the given angle that is
  between 0º and 360º or 0 radians and 2p radians.
  Note the quadrant of the coterminal angle.
• If the coterminal angle is in Quadrant I, do
  nothing. The coterminal angle is your reference
  angle.
• If the coterminal angle is in Quadrant II,
  subtract it from 180º or p radians.
• If the coterminal angle is in Quadrant III,
  subtract 180º or p radians from the angle.
• If the coterminal angle is in Quadrant IV,
  subtract it from 360º or 2p radians.

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Examples

• Find reference angles for each of the following