Difference between revisions of "Unit Circle - Essential Trigonometric Values"
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Finally, each '''<font color=#0033CC>blue</font>''' 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. | Finally, each '''<font color=#0033CC>blue</font>''' 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. | + | 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. |
| − | 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'''. | 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'''. | ||
Revision as of 08:21, 12 July 2015
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis, shown as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0^{\circ}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} radians, and each ordered pair, such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,0)} , 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} , and the other of absolute value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} . These are all a multiple of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/2} .
On the other hand, each point in a diagonal direction (shown in green) has a coordinate pair which are both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}/2} in absolute value. These are all an odd multiple of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/4} .
Finally, each blue direction has coordinates with two different absolute values, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{3}/2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2} . These fill out the remaining multiples of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/6} , or those which are not already shown in red.
It is important to realize that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{3}/2\,>\,1/2} , so whichever direction (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} ) seems smaller will take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2} as the absolute value of its coordinate.
As an example, consider the angle/direction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/6} . If we first consider the green diagonal at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/4} , where both coordinates share the same absolute value, we can see that the point on the unit circle at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/6} is to the right and below the diagonal. Thus, its Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} coordinate is bigger than its Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} coordinate, as can be seen in the image.
Also, recall that adding or subtracting the measure of a circle - either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 360^{\circ}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} - does NOT change the direction we are heading. As such, the angles Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2\pi,\,0,\,2\pi} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\pi} all point in the direction of the positive Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis. We may call such angles coterminal.