Trigonometric Functions: Unit Circle Approach

1
Trigonometric Functions Unit Circle Approach

• 5.2

2
Trigonometric Functions

• Sine (?) opp/hyp cosecant ? hyp/opp
• Cosine (?) adj/hyp secant ? hyp/adj
• Tangent (?) opp/adj cotangent ? adj/opp

3
Trigonometric Functions of 45?

• Sin (45) v(2)/2 csc 45 v2
• Cos (45) v(2)/2 sec 45 v2
• Tan (45) 1 cot 45 1

v2
45
1
45
1
4
Trigonometric Functions of 30?

• Sin (30) ½ csc 30 2
• Cos (30) /2 sec 30 2 /3
• Tan (30) /3 cot 30

2
30
60
1
5
Trigonometric Functions of 60?

• Sin (60) v3/2 csc 60 2 /3
• Cos (60) 1/2 sec 60 2
• Tan (60) v3 cot 60 /3

2
30
60
1
6
30, 45, and 60 degrees in Radians

• 30 degrees
• p/6 radians
• 45 degrees
• p/4 radians
• 60 degrees
• p/3 radians

7
Trig Functions on the Calculator

• Make sure the mode is radian/degree depending on
  your need.
• Press mode
• Arrow down to radian/degree and hit enter on the
  one wanted.
• There are no buttons on the calculator for sec,
  csc, or cot.
• Use 1/(the appropriate trig) to get these.
• On the calculator, the cube of sec 30 needs to be
  put in as (sec (30))3

8
Trig Functions on the Calculator

• Watch parenthesis.

9
Unit CircleA circle of radius one whose center
is on the origin.
(0,1)
(1,0)
(-1,0)
(0,-1)
10
Circle of Radius One

• Unit Circle
• Cos(angle) x-value
• Sin(angle)y-value
• If you have a circle with radius 1 and draw a
  triangle within it, opposite over hypotenuse will
  end up y-value over 1.

11
Relationships between Angles in the Four
Quadrants in degrees

• Sin (180 ?) sin(?)
• Sin (135) sin (45)
• Cos (180 ?) -cos(?)
• Cos(135) -cos(45)
• Cos(?) cos(- ?)
• Cos(45 ) cos(-45)
• Cos(-135) cos(135)
• Sin(- ?) -sin(?)
• Sin(-45) -sin(45)
• Sin(-135) -sin(135)

Remember that clockwise 45 degrees ( or -45) is
the same as counterclockwise 315 degrees. Also
clockwise 135 degrees (or -135) is the same as
counterclockwise 225 degrees.
12
Identities

• sin(180 - A) sin(A)
• cos(180 - A) -cos(A)

13
Sine is Odd, Cosine is Even

• sin(-A) -sin(A)
• cos(-A) cos(A)

14
Trig Functions of any Angle

• Standard Positionan angle whose vertex is at the
  origin and the initial side lies along the
  positive x-axis.
• Coterminal AnglesAngles with the same initial
  and terminal side, but different measures.
• Example 90, 450, and -270 are all coterminal
• They are angles that differ by an integer
  multiple of 360 degrees or an integer multiple of
  2p radians.

15
Circle of Radius One

• Unit Circle
• Cos(angle) x-value
• Sin(angle)y-value
• If you have a circle with radius 1 and draw a
  triangle within it, opposite over hypotenuse will
  end up y-value over 1.

16
Trigonometric Functions of any Angle

• Find the six trig functions of a given angle.
• Draw the angle in standard position.
• Create the reference triangle. This triangle
  must have the terminal side of the angle as its
  hypotenuse, and the vertical side must be
  perpendicular to and connect to the x-axis.
  Think Bowtie.
• Determine whether the triangle is a 30-60-90 or
  45-45-90.
• The values of the trig functions will equal those
  in the first quadrant only the signs might change.

17
The Bowtie
?
?
?
?
18
What is positive where?
All
Sin(csc)
Tan(cot)
Cos (sec)
19
What is positive where?
All
sin
tan
cos
20
Quadrantal Angles

• Angles whose terminal sides lie along one of the
  coordinate axis.
• They do not produce reference triangles. Just
  remember that x is the horizontal distance or
  zero if there isnt one. Y is the vertical
  distance, or zero if there isnt one.
• X and Y are either 0,1, or -1.

21
Quadrantal Angles in the Unit Circle
(0,1)
(1,0)
(-1,0)
(0,-1)
22
In the Unit Circlet is any real number

• Sin t y
• Cos t x
• Tan t (sin t)/(cos t) y/x
• Csc t 1/y (y not 0)
• Sec t 1/x (x not 0)
• Cot t (cos t)/(sin t) x/y