The Unit Circle - Simple Trigonometry
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The Unit Circle - Simple Trigonometry.
Trigonometry is the study of angles and the physical relationships between angles and geometry. To start, we use the unit circle, which is a circle of radius 1 unit, centered at the origin.
Part 1: Finding coordinates of points on the unit circle.
Four of the coordinates are easy: (1,0), (0,1), (-1,0) and (0,-1). These lie on the x and y axes and are called cardinal points (i.e. like the points on a compass). Draw any ray from the origin intersecting the circle. Call the intersection point P. This ray has an angle of theta "(". Therefore, the coordinates of point P are a function of the angle (. We make the following definition:
Given point P on the unit circle. Point P has coordinates (x,y).
x = cos (, and y = sin (.
Simple geometry (right triangles and pythagoras' formula) will help locate values for some angles such as 30(, 45( and 60(. Using symmetry on the unit circle you can find values for angles that are multiples of 30(, 45( and 60(. Your calculator will assist you for other angles.
First quadrant values for cos ( and sin (. That means 0 < ( < 90 (degrees) or 0 < ( < (/2 (radians).
Fill in the following table based on the figure above. Give exact values (no decimals).
|Angle ( in degrees |Angle ( in radians |x = cos ( |y = sin ( |
|A. 0 |0 |1 |0 |
|B. 30 |(/6 |[pic] |[pic] |
|C. 45 |(/4 |[pic] |[pic] |
|D. 60 |(/3 |[pic] |[pic] |
|E. 90 |(/2 |0 |1 |
1. What is the distance from point B to the origin? Why?
The distance from point B to the origin is the radius of the circle = 1.
2. A ray has an angle of ( = 20(. Find the point's coordinates.
x=cos 20(=.9397 y=sin 20(=.34202
3. What would the coordinates of P be for an angle of 390(? Why?
Since 390(=360(+30( an angle of 390( corresponds to a complete revolution plus 30(.
Thus the coordinates of P = (cos 390(,sin 390()=(cos 30(,sin 30()=([pic] )
Part II: Values of sin ( and cos ( for the other quadrants.
Using the table in Part I and appropriate symmetric points on the unit circle complete the following table. Some patterns should be evident. Give exact values. No decimals.
|Angle in degrees |Angle in radians |Which quadrant? |x = cos ( |y = sin ( |
|120 |2(/3 |2nd |([pic] |[pic] |
|135 |3(/4 |2nd |([pic] |[pic] |
|150 |5(/6 |2nd |([pic] |[pic] |
|210 |7(/6 |3rd |([pic] |([pic] |
|225 |5(/4 |3rd |([pic] |([pic] |
|240 |4(/3 |3rd |([pic] |([pic] |
|300 |5(/3 |4th |[pic] |([pic] |
|315 |7(/4 |4th |[pic] |([pic] |
|330 |11(/6 |4th |[pic] |([pic][pic] |
1. In quadrant II, the cos ( is always ___negative_ and the sin ( is always positive .
2. In quadrant III, the cos ( is always ___negative____, and the sin ( is always __negative____.
3. In quadrant IV, the cos ( is always __positive___, and the sin ( is always ___negative__.
4. Look at your results for the angles 135(, 225( and 315(, and compare them to the results for the angle of 45( from Quadrant I. What do you notice?
The values of sine and cosine are the same in absolute value but the signs are different according to which quadrant the angle is.
(The angle 45( is called the reference angle for the angles 135(, 225( and 315(.
5. What is the reference angle for 120(?
The reference angle is 180(-120(=60(
6. What is the reference angle for 335(?
The reference angle is 360(-335(=25(.
7. In general, if an angle ( is in the second quadrant, how would you find its reference angle?
The reference angle is the positive smallest angle formed by the terminal side of ( and the x-axis.
If 90( < ( ................
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