Trigonometric Functions of General Angles

1
Trigonometric Functions of General Angles

• Unit-5 Chapter-13 Section-3
• William Marinos

2
What You Will Learn

• Find values of trigonometric functions for
  general angles
• Use reference angles to find values of
  trigonometric functions

3
Vocab Review

• Initial side A fixed ray along the positive
  x-axis
• Terminal side A ray that rotates around the
  center
• Standard position When an angles is positioned
  so that its vertex is at the origin and its
  initial side is along the positive x-axis

4
Key Concept

• Let be an angle in standard position and
  let P(x,y) be a point on the terminal side of
  . Using the Pythagorean Theorem, the distance r
  from the origin to P is given by r
  . The trigonometric functions of an angle in
  standard position may be defined as follows

• Sin
• Cos
• Tan
• Csc
• Sec
• Cot

5
Examples 1 (pg. 717)

• Find the exact values of the six trigonometric
  functions of if the terminal side of
  contains the point (5,12).
• From the coordinates give, you know that x 5
  and y -12. Use the Pythagorean Theorem to find
  r.
• Pythagorean theorem
• Replace x with 5 and y with
  -12.
• or 13
• Now use x 5, y -12, and r 13 to write the
  ratios.

6
Quadrantal Angle

• If the terminal side of angle lies on one of
  the axes,
• is called a quadrantal angle. The quadrantal
  angles are 0, 90, 180, and 270 degrees. Either X
  or Y will equal 0. Since division by zero is
  undefined, two of the trigonometric values are
  undefined for each quadrantal angle.

7
Example 2

• Find the values of the six trigonometric
  functions for an angle in standard position that
  measures 270 degrees
• When 270, x 0 and y -r

8
Reference Angles

• To find the values of trig functions of angles
  greater than 90 degrees (or less than 0 degrees),
  you need to know how to find the measures of
  reference angles. If is a nonquadrantal
  angle in standard position, its reference angle,
  , is defined as the acute angle formed by
  the terminal side of and the x-axis
• The following rule can be used to find the
  reference angle for any nonquadrantal angle
  where 0 degrees lt lt 360 degrees (or 0 lt
  lt2

9
THE END