Trigonometry

1
Trigonometry

• Circular Functions
• By Miller Wright

2
Objective

• Further generalize the trigonometric functions by
  defining them in terms of the unit circle.

3
Recollection

• From Lesson 13-3
• cos ? y/r and sin ? x/r.
• If we are only working with the unit circle then
  r always equals 1. Therefore, cos ? y and sin ?
  x.

4
Definitions and Formulas

• Sine opposite / hypotenuse
• Cosine adjacent / hypotenuse
• Tangent opposite / adjacent
• Sine? Y
• Cosine? X

5
Solving For A Point, P

• The terminal side of the angle, ?, intersects the
  unit circle at a unique point, P(x,y). Recall
  that sin ? y/radius and cos ? x/radius. Since
  P(x,y) is on a unit circle where r 1, sin ?y
  and cos ?x.
• If the terminal side of an angle ? in standard
  position intersects the unit circle at P(x,y),
  then cos ? x and sin ? y. Therefore, the
  coordinates of P can be written as P(cos ?, sin
  ?).

(0,1)

P(co?,sn ?)
?
(1,0)
(-1,0)
(0,-1)
6
Circular Functions

• Since there is one point P for any angle, the
  relations cos ?x and sin ?y are functions of ?.
  Because they are both based on a unit circle they
  are called circular functions.

?
Theta
7
Periodic Functions
Cosine Curve
Sine Curve
8
Periodic Functions

• A function is called periodic if there is a
  number a such that f(x)f(xa) for all x in the
  domain of the function. The least positive value
  of a for which f(x)f(xa) is called the period
  of the function.