Trigonometric Functions of Any Angle & Polar Coordinates

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Trigonometric Functions of Any AnglePolar
Coordinates
Sections 8.1, 8.2, 8.3, 21.10
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Definitions of Trig Functions of Any Angle (Sect
8.1)
Definitions of Trigonometric Functions of Any
Angle Let ? be an angle in standard position with
(x, y) a point on the terminal side of ? and

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The Signs of the Trig Functions
Since the radius is always positive (r gt 0), the
signs of the trig functions are dependent upon
the signs of x and y. Therefore, we can
determine the sign of the functions by knowing
the quadrant in which the terminal side of the
angle lies.
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The Signs of the Trig Functions
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Where each trig function is POSITIVE

All Students Take Calculus
Translation A All 3 functions are positive in
Quad 1 S Sine function is positive in Quad 2 T
Tangent function is positive in Quad 3 C Cosine
function is positive in Quad 4
In Quad 2, sine is positive, but cosine and
tangent are negative in Quad 3, tangent is
positive, but sine and cosine are negative in
Quad 4, cosine is positive but sine and tangent
are negative. Reciprocal functions have the
same sign. So cosecant is positive wherever sine
is positive, secant is positive wherever cosine
is positive, and cotangent is positive wherever
tangent is positive.
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Example
Determine if the following functions are positive
or negative
sin 210
cos 320
cot (-135)
csc 500
tan 315
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Examples

• For the given values, determine the quadrant(s)
  in which the terminal side of ? lies.

8
Examples

• Determine the quadrant in which the terminal side
  of ? lies, subject to both given conditions.

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Examples

• Find the exact value of the six trigonometric
  functions of ? if the terminal side of ? passes
  through point (3, -5).

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Reference Angles (Sect 8.2)
The values of the trig functions for non-acute
angles (Quads II, III, IV) can be found using the
values of the corresponding reference angles.
Definition of Reference Angle Let ? be an angle
in standard position. Its reference angle is
the acute angle formed by the terminal side of ?
and the horizontal axis.
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Example
Find the reference angle for
Solution
By sketching ? in standard position, we see that
it is a 3rd quadrant angle. To find , you
would subtract 180 from 225 .
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• So whats so great about reference angles?
• Wellto find the value of the trig function of
  any non-acute angle, we just need to find the
  trig function of the reference angle and then
  determine whether it is positive or negative,
  depending upon the quadrant in which the angle
  lies.
• For example,

45 is the ref angle
In Quad 3, sin is negative
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Example

• Give the exact value of the trig function
  (without using a calculator).
• 
  

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Examples (Text p 239 6 8)

• Express the given trigonometric function in terms
  of the same function of a positive acute angle.
• 
  

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• Now, of course you can simply use the calculator
  to find the value of the trig function of any
  angle and it will correctly return the answer
  with the correct sign.
• Remember
• Make sure the Mode setting is set to the correct
  form of the angle Radian or Degree
• To find the trig functions of the reciprocal
  functions (csc, sec, and cot), use the ? button
  or enter ??original function .

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Example

• Evaluate . Round appropriately.
• Set Mode to Degree
• Enter ?????? OR ???????

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HOWEVER, it is very important to know how to use
the reference angle when we are using the inverse
trig functions on the calculator to find the
angle because the calculator may not directly
give you the angle you want.
Example Find the value of ? to the nearest
0.01
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Examples
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Examples
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BONUS PROBLEM
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SUPER DUPER BONUS PROBLEM
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Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles
whose terminal side falls on one of the axes

, we will use the circle.
(0, 1)

• Unit Circle
• Center (0, 0)
• radius 1
• x2 y2 1

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Now using the definitions of the trig functions
with r 1, we have
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Example
Find the value of the six trig functions for
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Example
Find the value of the six trig functions for
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Example
Find the value of the six trig functions for
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Radian Measure (Sect 8.3)
A second way to measure angles is in radians.
Definition of Radian One radian is the measure
of a central angle ? that intercepts arc s equal
in length to the radius r of the circle.
In general, for ? in radians,
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Radian Measure
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Radian Measure
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Conversions Between Degrees and Radians
Example Convert from degrees to radians 210º
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Conversions Between Degrees and Radians
Example a) Convert from radians to degrees
b) Convert from radians to degrees 3.8
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Conversions Between Degrees and Radians
c) Convert from degrees to radians (exact)
d) Convert from radians to degrees
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Conversions Between Degrees and Radians
Again! e) Convert from degrees to radians (to 3
decimal places) f) Convert from radians
to degrees (to nearest tenth) 1 rad
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Examples
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Polar Coordinates (Sect 21.10)

• A point P in the polar coordinate system is
  represented by an ordered pair .
• If , then r is the distance of the point
  from the pole.
• ? is an angle (in degrees or radians) formed by
  the polar axis and a ray from the pole through
  the point.

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Polar Coordinates
If , then the point is located units
on the ray that extends in the opposite direction
of the terminal side of ?.
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Example Plot the point P with polar coordinates
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• Example

Plot the point with polar coordinates
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Plotting Points Using Polar Coordinates
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Plotting Points Using Polar Coordinates
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A)
B)
D)
C)
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To find the rectangular coordinates for a point
given its polar coordinates, we can use the trig
functions.
Example
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Likewise, we can find the polar coordinates if we
are given the rectangular coordinates using the
trig functions.
Example Find the polar coordinates (r, ?) for
the point for which the rectangular coordinates
are (5, 4). Express r and ? (in radians) to three
sig digits.
(5, 4)
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Conversion from Rectangular Coordinates to Polar
Coordinates If P is a point with rectangular
coordinates (x, y), the polar coordinates (r, ?)
of P are given by
You need to consider the quadrant in which P
lies in order to find the value of ?.
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Find polar coordinates of a point whose
rectangular coordinates are given. Give exact
answers with ? in degrees.
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Find polar coordinates of a point whose
rectangular coordinates are given. Give exact
answers with ? in degrees.
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The TI-84 calculator has handy conversion
features built-in. Check out the ANGLE menu.
5 Returns value of r given rectangular
coordinates (x, y) 6 Returns value of ? given
rectangular coordinates (x, y) 7 Returns value
of x given polar coordinates (r, ?) 8 Returns
value of y given polar coordinates (r, ?)
Check the MODE for the appropriate setting for
angle measure (degrees vs. radians).
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• End of Section