Difference between revisions of "Unit Circle - Essential Trigonometric Values"

From Math Wiki
Jump to navigation Jump to search
Line 2: Line 2:
  
 
== Related Pairs for Sine and Cosine ==
 
== Related Pairs for Sine and Cosine ==
Defined by the equation <math style="vertical-align: -4px">x^2+y^2\,=\,1</math>, the unit circle is the collection of points that lie one unit from the origin. For trigonometry, it relates directions, called out in degrees or radians, to their cosine and sine.  We measure angles counterclockwise from the positive <math style="vertical-align: 0px">x</math>-axis, shown as <math style="vertical-align: 0px">0^{\circ}</math> or <math style="vertical-align: 0px">0</math> radians.  Each ordered pair, such as <math style="vertical-align: -5px">(1,0)</math>, is both a point on the unit circle <u style="vertical-align: 0px">''and''</u> the cosine and sine in that direction or angle.  In the image, we have color coded points and directions which are related through absolute values.
+
Defined by the equation <math style="vertical-align: -4px">x^2+y^2\,=\,1</math>, the unit circle is the collection of points that lie one unit from the origin. For trigonometry, it relates directions, called out in degrees or radians, to their cosine and sine.  We measure angles counterclockwise from the positive <math style="vertical-align: 0px">x</math>-axis, shown as <math style="vertical-align: 0px">0^{\circ}</math> or <math style="vertical-align: 0px">0</math> radians, and each ordered pair, such as <math style="vertical-align: -5px">(1,0)</math>, is both a point on the unit circle <u style="vertical-align: 0px">''and''</u> the cosine and sine in that direction or angle.  In the image, we have color coded points and directions which are related through absolute values.
  
 
Each red point of the unit circle lies on an axis, and has one coordinate of <math style="vertical-align: 0px">0</math>, and the other of absolute value <math style="vertical-align: -1px">1</math>. These are all a multiple of <math style="vertical-align: -5px">\pi/2</math>.
 
Each red point of the unit circle lies on an axis, and has one coordinate of <math style="vertical-align: 0px">0</math>, and the other of absolute value <math style="vertical-align: -1px">1</math>. These are all a multiple of <math style="vertical-align: -5px">\pi/2</math>.

Revision as of 07:52, 12 July 2015

UnitCircle.png

Related Pairs for Sine and Cosine

Defined by the equation , the unit circle is the collection of points that lie one unit from the origin. For trigonometry, it relates directions, called out in degrees or radians, to their cosine and sine. We measure angles counterclockwise from the positive -axis, shown as or radians, and each ordered pair, such as , is both a point on the unit circle and the cosine and sine in that direction or angle. In the image, we have color coded points and directions which are related through absolute values.

Each red point of the unit circle lies on an axis, and has one coordinate of , and the other of absolute value . These are all a multiple of .

On the other hand, each point in a diagonal direction (shown in green) has a coordinate pair which are both in absolute value. These are all an odd multiple of .

Finally, each blue direction has coordinates with two different absolute values,   and . These fill out the remaining multiples of , or those which are not already shown in red.

It is important to realize that , so whichever direction ( or ) seems smaller will take as the absolute value of its coordinate.

As an example, consider the angle/direction . If we first consider the green diagonal at , where both coordinates share the same absolute value, we can see that the point on the unit circle at is to the right and below the diagonal. Thus, its coordinate is bigger than its coordinate, as can be seen in the image.

Also, recall that adding or subtracting the measure of a circle - either or - does NOT change the direction we are heading. As such, the angles and all point in the direction of the positive -axis. We may call such angles coterminal.