Unit Circle - Essential Trigonometric Values - Revision history

MathAdmin: /* Related Pairs for Sine and Cosine */

2015-07-14T05:08:25Z

Related Pairs for Sine and Cosine

MathAdmin: /* Related Pairs for Sine and Cosine */

2015-07-12T16:23:54Z

Related Pairs for Sine and Cosine

MathAdmin: /* Related Pairs for Sine and Cosine */

2015-07-12T16:21:17Z

Related Pairs for Sine and Cosine

MathAdmin: /* Related Pairs for Sine and Cosine */

2015-07-12T15:14:30Z

Related Pairs for Sine and Cosine

MathAdmin: /* Related Pairs for Sine and Cosine */

2015-07-12T14:52:58Z

Related Pairs for Sine and Cosine

MathAdmin: /* Related Pairs for Sine and Cosine */

2015-07-12T14:52:01Z

Related Pairs for Sine and Cosine

MathAdmin: Created page with "900px == Related Pairs for Sine and Cosine == In the image, we have color coded points and directions which are related through absolute values..."

2015-07-12T05:51:02Z

Created page with "center|900px == Related Pairs for Sine and Cosine == In the image, we have color coded points and directions which are related through absolute values..."

New page

[[File:UnitCircle.png|center|900px]]

== Related Pairs for Sine and Cosine ==
In the image, we have color coded points and directions which are related through absolute values. We call out angles by measuring 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.

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>.

On the other hand, each point in a diagonal direction (shown in green) has a coordinate pair which are both <math style="vertical-align: -5px">\sqrt{2}/2</math> in absolute value. These are all an odd multiple of <math style="vertical-align: -5px">\pi/4</math>.

Finally, each blue direction has coordinates with two different absolute values, <math style="vertical-align: -5px">\sqrt{3}/2</math>  and <math style="vertical-align: -5px">1/2</math>. These fill out the remaining multiples of <math style="vertical-align: -5px">\pi/6</math>, or those which are not already shown in red.

It is important to realize that <math style="vertical-align: -5px">\sqrt{3}/2\,>\,1/2</math>, so whichever direction (<math style="vertical-align: 0px">x</math> or <math style="vertical-align: -4px">y</math>) seems smaller will take <math style="vertical-align: -5px">1/2</math> as the absolute value of its coordinate. Consider the angle/direction <math style="vertical-align: -5px">\pi/6</math>. If we first consider the green diagonal at <math style="vertical-align: -5px">\pi/4</math>, where both coordinates share the same absolute value, we can see that the point on the unit circle at <math style="vertical-align: -5px">\pi/6</math> is to the right and below the diagonal. Thus, its <math style="vertical-align: 0px">x</math> coordinate is bigger than its <math style="vertical-align: -4px">y</math> coordinate, as can be seen in the image.

Also, recall that adding or subtracting the measure of a circle - either <math style="vertical-align: 0px">360^{\circ}</math> or <math style="vertical-align: 0px"> 2\pi</math> - does NOT change the direction we are heading. For example, the angles <math style="vertical-align: -4px">-22\pi,\,0,\,2\pi</math> and <math style="vertical-align: -1px">4\pi</math> all point in the direction of the positive <math style="vertical-align: 0px">x</math>-axis.

@@ Line 12: / Line 12: @@
 It is important to realize that <math style="vertical-align: -5px">\sqrt{3}/2\,>\,1/2</math>, so whichever direction (<math style="vertical-align: 0px">x</math> or <math style="vertical-align: -4px">y</math>) seems smaller will take <math style="vertical-align: -5px">1/2</math>&thinsp; as the absolute value of its coordinate.
-As an example, consider the angle/direction <math style="vertical-align: -5px">\pi/6</math>.  If we first consider the green diagonal at <math style="vertical-align: -5px">\pi/4</math>, where both coordinates share the same absolute value, we can see that the point on the unit circle at <math style="vertical-align: -5px">\pi/6</math> is to the right and below the diagonal.  Thus, its <math style="vertical-align: 0px">x</math> coordinate is bigger than its <math style="vertical-align: -4px">y</math> coordinate, as can be seen in the image.
+As an example, consider the angle/direction <math style="vertical-align: -5px">\pi/6</math>.  If we first consider the green diagonal at <math style="vertical-align: -5px">\pi/4</math>, where both coordinates share the same absolute value, we can see that the point on the unit circle at <math style="vertical-align: -5px">\pi/6</math>&thinsp; is to the right and below the diagonal.  Thus, its <math style="vertical-align: 0px">x</math> coordinate is bigger than its <math style="vertical-align: -4px">y</math> coordinate, as can be seen in the image.
-Also, recall that adding or subtracting the measure of a circle - either <math style="vertical-align: 0px">360^{\circ}</math> or <math style="vertical-align: 0px"> 2\pi</math> - does NOT change the direction we are heading.  As such, the angles <math style="vertical-align: -4px">-2\pi,\,0,\,2\pi</math> and <math style="vertical-align: -1px">4\pi</math> all point in the direction of the positive <math style="vertical-align: 0px">x</math>-axis.   We may call such angles '''coterminal'''.
+Also, recall that adding or subtracting the angular measure of a circle - either <math style="vertical-align: 0px">360^{\circ}</math> or <math style="vertical-align: 0px"> 2\pi</math> - does ''<u>not</u>'' change the direction we are heading.  As such, the angles <math style="vertical-align: -4px">-2\pi,\,0,\,2\pi</math> and <math style="vertical-align: -1px">4\pi</math> all point in the direction of the positive <math style="vertical-align: 0px">x</math>-axis.   We call such angles '''coterminal'''.

@@ Line 12: / Line 12: @@
 It is important to realize that <math style="vertical-align: -5px">\sqrt{3}/2\,>\,1/2</math>, so whichever direction (<math style="vertical-align: 0px">x</math> or <math style="vertical-align: -4px">y</math>) seems smaller will take <math style="vertical-align: -5px">1/2</math>&thinsp; as the absolute value of its coordinate.
-As an example, consider the angle/direction <math style="vertical-align: -5px">\pi/6</math>.  If we first consider the green diagonal at <math style="vertical-align: -5px">\pi/4</math>, where both coordinates share the same absolute value, we can see that the point on the unit circle at <math style="vertical-align: -5.5px">\pi/6</math> is to the right and below the diagonal.  Thus, its <math style="vertical-align: 0px">x</math> coordinate is bigger than its <math style="vertical-align: -4px">y</math> coordinate, as can be seen in the image.
+As an example, consider the angle/direction <math style="vertical-align: -5px">\pi/6</math>.  If we first consider the green diagonal at <math style="vertical-align: -5px">\pi/4</math>, where both coordinates share the same absolute value, we can see that the point on the unit circle at <math style="vertical-align: -5px">\pi/6</math> is to the right and below the diagonal.  Thus, its <math style="vertical-align: 0px">x</math> coordinate is bigger than its <math style="vertical-align: -4px">y</math> coordinate, as can be seen in the image.
 Also, recall that adding or subtracting the measure of a circle - either <math style="vertical-align: 0px">360^{\circ}</math> or <math style="vertical-align: 0px"> 2\pi</math> - does NOT change the direction we are heading.  As such, the angles <math style="vertical-align: -4px">-2\pi,\,0,\,2\pi</math> and <math style="vertical-align: -1px">4\pi</math> all point in the direction of the positive <math style="vertical-align: 0px">x</math>-axis.   We may call such angles '''coterminal'''.

@@ Line 10: / Line 10: @@
 Finally, each '''<font color=#0033CC>blue</font>''' direction has coordinates with two different absolute values, <math style="vertical-align: -5px">\sqrt{3}/2</math>&thinsp; and <math style="vertical-align: -5px">1/2</math>.  These fill out the remaining multiples of <math style="vertical-align: -5px">\pi/6</math>, or those which are not already shown in red.
-It is important to realize that <math style="vertical-align: -5px">\sqrt{3}/2\,>\,1/2</math>, so whichever direction (<math style="vertical-align: 0px">x</math> or <math style="vertical-align: -4px">y</math>) seems smaller will take <math style="vertical-align: -5px">1/2</math> as the absolute value of its coordinate.
+It is important to realize that <math style="vertical-align: -5px">\sqrt{3}/2\,>\,1/2</math>, so whichever direction (<math style="vertical-align: 0px">x</math> or <math style="vertical-align: -4px">y</math>) seems smaller will take <math style="vertical-align: -5px">1/2</math>&thinsp; as the absolute value of its coordinate.
-As an example, consider the angle/direction <math style="vertical-align: -5px">\pi/6</math>.  If we first consider the green diagonal at <math style="vertical-align: -5px">\pi/4</math>, where both coordinates share the same absolute value, we can see that the point on the unit circle at <math style="vertical-align: -5px">\pi/6</math> is to the right and below the diagonal.  Thus, its <math style="vertical-align: 0px">x</math> coordinate is bigger than its <math style="vertical-align: -4px">y</math> coordinate, as can be seen in the image.
+As an example, consider the angle/direction <math style="vertical-align: -5px">\pi/6</math>.  If we first consider the green diagonal at <math style="vertical-align: -5px">\pi/4</math>, where both coordinates share the same absolute value, we can see that the point on the unit circle at <math style="vertical-align: -5.5px">\pi/6</math> is to the right and below the diagonal.  Thus, its <math style="vertical-align: 0px">x</math> coordinate is bigger than its <math style="vertical-align: -4px">y</math> coordinate, as can be seen in the image.
 Also, recall that adding or subtracting the measure of a circle - either <math style="vertical-align: 0px">360^{\circ}</math> or <math style="vertical-align: 0px"> 2\pi</math> - does NOT change the direction we are heading.  As such, the angles <math style="vertical-align: -4px">-2\pi,\,0,\,2\pi</math> and <math style="vertical-align: -1px">4\pi</math> all point in the direction of the positive <math style="vertical-align: 0px">x</math>-axis.   We may call such angles '''coterminal'''.

@@ Line 4: / Line 4: @@
 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 '''<font color=#FF0000>red</font>''' 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>.
-On the other hand, each point in a diagonal direction (shown in green) has a coordinate pair which are both <math style="vertical-align: -5px">\sqrt{2}/2</math> in absolute value.  These are all an odd multiple of <math style="vertical-align: -5px">\pi/4</math>.
+On the other hand, each point in a diagonal direction (shown in '''<font color=#009933>green</font>''') has a coordinate pair which are both <math style="vertical-align: -5px">\sqrt{2}/2</math> in absolute value.  These are all an odd multiple of <math style="vertical-align: -5px">\pi/4</math>.
-Finally, each blue direction has coordinates with two different absolute values, <math style="vertical-align: -5px">\sqrt{3}/2</math>&thinsp; and <math style="vertical-align: -5px">1/2</math>.  These fill out the remaining multiples of <math style="vertical-align: -5px">\pi/6</math>, or those which are not already shown in red.
+Finally, each '''<font color=#0033CC>blue</font>''' direction has coordinates with two different absolute values, <math style="vertical-align: -5px">\sqrt{3}/2</math>&thinsp; and <math style="vertical-align: -5px">1/2</math>.  These fill out the remaining multiples of <math style="vertical-align: -5px">\pi/6</math>, or those which are not already shown in red.
 It is important to realize that <math style="vertical-align: -5px">\sqrt{3}/2\,>\,1/2</math>, so whichever direction (<math style="vertical-align: 0px">x</math> or <math style="vertical-align: -4px">y</math>) seems smaller will take <math style="vertical-align: -5px">1/2</math> as the absolute value of its coordinate.

@@ Line 2: / Line 2: @@
 == 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>.