Download Trigonometric Functions : Unit Circle Approach

Trigonometric Functions : Unit Circle
Approach
08/29/2012
Unit Cirlce: circle with radius 1 and center is at the origin of a rectangular
coordinate system.
If we have circle with radius r then circumference of that circle is 2 πr =⇒
circumference of unit circle = 2π
1 revolution around unit cirlce =⇒ the length of arc is 2π units.
Note: Let t be any real number. Position t-axis so that it is vertical with
the positive direction up. Place this t-axis in the xy-plan so that t=0 is
located at point (1,0) in the xy-plane. If t ≥ 0, let s be the distance from the
origin to t on the t-axis. Now take a unit circle. Beginning at the point (1,0)
on the unit cirlce, travel s=t units in the counterclockwise direction along
the unit cirlce, to arrive at ehe point P=(x,y). If t0, begin at the point (1,0)
on the unit cirlce and travel s=—t— units in clockwise dircetion to arrive at
the point P=(x,y). So for any real number t, we can locate a unique point
P=(x,y). We call P the point on the unit circle that corresponds to
t.
Definition: Let t be real number and P=(x,y) be the point on the unit
circle that corresponds to t.
The sine function associates with t the y-coordinate of P and is denoted
by
sint = y
The cosine function: cost = x
If x 6= 0, the tangent function: tant = xy and secant function: sect = x1
If y 6= 0, the cosecant function: csct = y1 and cotangent function:
cott = xy
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Definition: If θ = t radians, the six trigonometric functions of the angle θ
are :
sinθ=sint, cosθ = cost, tanθ = tant
cscθ = csct, secθ = sect, cotθ = cott
Objectives:
1. Find the Exact Values of the trigonometric Functions using a point on
the unit cirlce.
2. Find the Exact values of the trigonometric functions of quadrantal
angles.
3. Find the exact values of trigonometric funtions of
π
4
= 450
4. Find the exact values of the trigonometric functions of
π
= 600
3
π
6
= 600 and
5. Find the exact values of the trigonometric funtions for integer multiples
of π6 = 300 , π4 = 450 , and π3 = 600 .
6. Use calculater to approximate the value of trigonometric function.
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