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Trigonometry

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Sine= opposite / hypotenuse. Cosine= adjacent / hypotenuse. Tangent= opposite / adjacent ... The terminal side of the angle, ?, intersects the unit circle at a ... – PowerPoint PPT presentation

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Title: 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.
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