Difference between revisions of "Trigonometric Functions: Unit Circle Approach"

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The Unit Circle is a circle with its center at the origin (0,0) and a radius of one unit.
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==The Unit Circle==
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Consider an angle in standard position, such that the point (''x'', ''y'') on the terminal side of the angle is a point on a circle with radius 1.
  
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[[File:Unit circle 3.svg|thumb|300px|Unit Circle]]
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:[[File:Defining Trigonometric Functions Figure 4.svg|500px]]
  
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Angles are always measured from the positive x-axis (also called the "right horizon").  Angles measured counterclockwise have ''positive'' values; angles measured clockwise have ''negative'' values.
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This circle is called the '''unit circle'''. With ''r'' = 1, we can define the trigonometric functions in the unit circle:
  
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{{clear}}
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{| style="background:transparent;" cellpadding="5"
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| style="padding-right:30px;" | <math>\cos \theta = \frac{x}{r} = \frac{x}{1} = x</math> || <math>\sec \theta = \frac{r}{x} = \frac{1}{x}</math>
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|-
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| <math>\sin \theta = \frac{y}{r} = \frac{y}{1} = y</math> || <math>\csc \theta = \frac{r}{y} = \frac{1}{y}</math>
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|-
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| <math>\tan \theta = \frac{y}{x}</math> || <math>\cot \theta = \frac{x}{y}</math>
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|}
  
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==Defining sine and cosine in terms of the unit circle==
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Notice that in the unit circle, the sine and cosine of an angle are the ''x'' and ''y'' coordinates of the point on the terminal side of the angle. Now we can find the values of the trigonometric functions of any angle of rotation, even the quadrantal angles, which are not angles in triangles.
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In the unit circle shown here, a unit-length radius has been drawn from the origin to a point (x,&nbsp;y) on the circle.
 
  
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[[Image:Defining_sine_and_cosine.svg|300px|thumb|center|Defining sine and cosine]]
 
  
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A line perpendicular to the x-axis, drawn through the point (x,&nbsp;y), intersects the x-axis at the point with the [[Algebra/Function Graphing|abscissa]] ''x''. Similarly, a line perpendicular to the y-axis intersects the y-axis at the point with the [[Algebra/Function Graphing|ordinate]] ''y''. The angle between the x-axis and the radius is <math>\alpha</math>.
 
  
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So, we can say that the sine of an angle is the ordinate of the point on the unit circle at that angle, and the cosine of an angle is the abscissa of the point on the unit circle at that angle.
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[[File:Unit Circle Definitions of Six Trigonometric Functions.png|The center of the unit circle is the origin <math>O</math>, point <math>A=(x_A,y_A)</math>. Creating line segments starting from point <math>A</math> to <math>x</math> on the <math>x</math>-axis will create two line segments. From this, we learn that, in terms of <math>\theta</math>, point <math>A=(\cos\theta,\sin\theta)</math>and other properties.]]
  
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We define the basic trigonometric functions of any angle <math>\alpha</math> as follows:
 
  
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{{TrigBoxOpen}}
 
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<math>
 
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\begin{matrix}
 
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\mathrm{sine:} & \sin(\alpha) & = & y \\
 
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\mathrm{cosine:} & \cos(\alpha) & = & x \\
 
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\end{matrix}
 
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</math>
 
  
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<math>\tan(\theta)</math> can be algebraically defined.
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The center of the unit circle is the origin <math>O</math>, point <math>A=(x_A,y_A)</math>. Creating line segments starting from point <math>A</math> to <math>x</math> on the <math>x</math>-axis will create two line segments. From this, we learn that, in terms of <math>\theta</math>, point <math>A=(\cos\theta,\sin\theta)</math>and other properties.
  
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<math>\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}\qquad\cos(\theta)\ne0</math>
 
  
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<math>\tan(\alpha)=\frac{y}{x}\qquad x\ne0</math>
 
  
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{{TrigBoxClose}}
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[[File:Unit_circle_angles.svg|Labeled unit circle]]
  
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These three trigonometric functions can be used whether the angle is measured in degrees or radians as long as it specified which, when calculating trigonometric functions from angles or vice versa.
 
  
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===Alternative definitions===
 
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* A previous chapter used ''[[Trigonometry/Soh-Cah-Toa|Soh-Cah-Toa]]'' to define the trigonometric functions. The advantage of the unit circle is that &theta; can be extended outside the first quadrant <math>\left[0,\frac{\pi}{2}\right]</math>, which allows us to define these functions on the interval <math>(-\infty,\infty)</math>.
 
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* If trigonometry is applied to vectors, it is more convenient, if the radius of the circle is not equal to unity. For example, if vector '''''A''''' has magnitude <math>A=\left|\mathbf{A}\right\vert</math>:
 
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{{TrigBoxOpen}}
 
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<math>
 
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\begin{matrix}
 
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x=r\cos(\alpha)&\to & \mathbf{A}_x=\mathbf{A}\cos(\alpha) \\
 
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y=r\sin(\alpha)&\to & \mathbf{A}_y=\mathbf{A}\sin(\alpha) \\
 
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r=\sqrt{x^2+y^2}&\to & \mathbf{A}=\sqrt{\left(\mathbf{A}_x\right)^{2}+\left(\mathbf{A}_y\right)^{2}}\\
 
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\end{matrix}
 
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</math>
 
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{{TrigBoxClose}}
 
  
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It is important to know why the above equations are true. Knowing <math>\cos(\alpha)=\frac{x}{r}</math>, <math>r\cdot\cos(\alpha)=r\cdot\frac{x}{r}=x</math>. The same could be said for the definition for <math>y</math>. Finally, the final line is the Pythagorean identity.
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We can use the figure above to determine values of the trig functions for the quadrantal angles. For example, <math>\sin{90^\circ} = y = 1</math>. <math>\sec{45^\circ} = \frac{1}{x} = \frac{2}{\sqrt{2}} = \sqrt{2}</math>.
  
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==Video links==
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===Example 1===
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More about this topic can be found at the '''Khan Academy'':'
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Use the unit circle above to find each value:
  
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* [http://www.khanacademy.org/video/the-unit-circle-definition-of-trigonometric-function?playlist=Trigonometry The unit circle definition of trigonometric function]
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a. cos 90°
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* [http://www.khanacademy.org/video/unit-circle-definition-of-trig-functions?playlist=Trigonometry Unit circle definition of trig functions]
 
  
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==Some values for sine and cosine==
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b. cot 180°
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A unit circle with certain exact values marked on it is below:
 
  
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[[File:Unit_circle_angles.svg|thumb|397x397px|left|Labeled unit circle]]
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c. sec 0°
  
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Unit circles form the basis of most analog clocks and animations on computers since the cos and sin correspond to the x and y positions of the end of the line segments representing the hands of the clock.
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----
  
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The unit circle on the left has the degree, the radian, and the coordinate value on the unit circle. For a coordinate value <math>(x,y)</math>, if walking around the circle <math>\frac{\pi}{3}</math> radians anti-clockwise from the horizontal axis, the coordinate value at which the person walked around the circle is <math>\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)</math>.
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'''Solution''':
  
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Remember that on a unit circle, the angle <math>\theta</math> anti-clockwise from the horizontal axis gives <math>\cos(\theta)=x</math> and <math>\sin(\theta)=y</math>. The same is true for radians. As such, for <math>(x,y)</math> corresponding to <math>(\cos(\theta),\sin(\theta))</math> on the unit circle, <math>\frac{\pi}{3}</math> radians when substituted in the cosine or sine function is the coordinate value on the unit circle. That is:
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a. cos 90° = 0
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:<math>\left(\cos\left(\frac{\pi}{3}\right),\sin\left(\frac{\pi}{3}\right)\right)\to\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)</math>, OR, equivalently,
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:The ordered pair for this angle is (0, 1). The cosine value is the ''x'' coordinate, 0.
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:<math>\left(\cos\left(60^{\circ}\right),\sin\left(60^{\circ}\right)\right)\to\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)</math>
 
  
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The unit circle is very useful to your mathematical studies of trigonometry because it tells you the EXACT value of certain angles. Later, you will learn how to find other ratios of angles and radians without needing to rely on the special values of the unit circle – 30, 45, 60, and 90.
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b. cot 180° is undefined
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: The ordered pair for this angle is (-1, 0). The ratio <math>\tfrac{x}{y}</math> is <math>\tfrac{-1}{0}</math>, which is undefined.
  
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It is worth your time memorizing some of the values of sine and cosine on the unit circle (cosine is equal to <math>x</math> while sine is equal to <math>y</math>). You should at least become familiar with the values for <math>0^\circ, 30^\circ 45^\circ, 90^\circ</math> and know where <math>\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi</math> are on the unit circle.
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c. sec 0° = 1
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: The ordered pair for this angle is (1, 0).  The ratio is <math>\tfrac{1}{x}</math> is <math>\tfrac{1}{1}</math> = 1.
  
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If you have some trouble memorizing the values, here are some helpful hints and patterns. Try to find some more other than what is listed.
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There are several important angles in the unit circle that you will work with extensively in your study of trigonometry: 30°, 45°, and 60°. To find the values of the trigonometric functions of these angles, we need to know the ordered pairs. Let's begin with 30°.
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* The coordinate value <math>(x,y)</math> on the unit circle has the <math>\text{numerator}</math> decrease from <math>\sqrt{3}</math> in the <math>x</math>-value coordinate at <math>\frac{\pi}{6}</math> to <math>\sqrt{1}=1</math> at <math>\frac{\pi}{3}</math>. The denominator is always <math>2</math>. Here, see what we mean. Ignore the y-value for now:
 
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**<math>\left(\frac{\sqrt{3}}{2},\sin\left(\frac{\pi}{6}\right)\right)</math>, <math>\left(\frac{\sqrt{2}}{2},\sin\left(\frac{\pi}{4}\right)\right)</math>, <math>\left(\frac{1}{2},\sin\left(\frac{\pi}{3}\right)\right)</math>.
 
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*Like the <math>x</math>-value coordinate, the <math>y</math>-value coordinate has a pattern in the numerator. As the angle increases, between <math>30^\circ</math> and <math>60^\circ</math> (inclusive), the <math>\text{numerator}</math> increases from <math>\sqrt{1}=1</math> in the <math>y</math>-value coordinate at <math>\frac{\pi}{6}</math> to <math>\sqrt{3}</math> at <math>\frac{\pi}{3}</math>. From the above bullet, putting it together, you get the following pattern:
 
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**<math>\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)</math>, <math>\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)</math>, <math>\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)</math>.
 
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* Certain radian values are simply reflections of the other. Take a look at the ones with the same denominator and see if you can correspond any patterns to what you see.
 
  
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If you are to ever be tested on this, make a quick sketch of the first quadrant of the circle, and remember the pattern that underlies the unit circle.
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The terminal side of the angle intersects the unit circle at the point <math>\tfrac{\sqrt{3}}{2}, \tfrac{1}{2}</math>. (You will prove this in one of
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{{clear}}
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the review exercises.). Therefore we can find the values of any of the trig functions of 30°. For example, the cosine value is the ''x''-coordinate, so cos (30°) = <math>\tfrac{\sqrt{3}}{2}</math>.  Because the coordinates are fractions, we have to do a bit more work in order to find the tangent value:
  
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Clock Hands:
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:<math>\tan (30^\circ) = \frac{y}{x} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{2} \cdot \frac{2}{\sqrt{3}} = \frac{1}{\sqrt{3}}</math>
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* What angle (a) in degrees and (b) in radians do the hour and minute hands of a clock move through in:
 
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* One hour
 
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* One minute
 
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* How far apart in degrees are the hour and minute hand at five minutes past five? (you should take into account that the hour hand is not exactly at the five, but has moved a little further)
 
  
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In the review exercises you will find the values of the remaining four trig functions of this angle. The table below summarizes the ordered pairs for 30°, 45°, and 60° on the unit circle.
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{| class="wikitable"
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|+ Table 1.10
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! scope="col" | Angle
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! scope="col" | ''x'' coordinate
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! scope="col" | ''y'' coordinate
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|-
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| 30° || <math>\tfrac{\sqrt{3}}{2}</math> || <math>\tfrac{1}{2}</math>
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|-
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| 45° || <math>\tfrac{\sqrt{2}}{2}</math> || <math>\tfrac{\sqrt{2}}{2}</math>
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|-
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| 60° || <math>\tfrac{1}{2}</math> || <math>\tfrac{\sqrt{3}}{2}</math>
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|}
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We can use these values to find the values of any of the six trig functions of these angles.
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===Example 2===
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Find the value of each function.
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a. cos (45°)
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b. sin (60°)
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c. tan (45°)
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----
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'''Solution''':
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a. cos (45°) = <math>\tfrac{\sqrt{2}}{2}</math>
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: The cosine value is the ''x'' coordinate of the point.
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b. sin (60°) = <math>\tfrac{\sqrt{3}}{2}</math>
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: The sine value is the ''y'' coordinate of the point.
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c. tan (45°) = 1
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: The tangent value is the ratio of the ''y'' coordinate to the ''x'' coordinate. Because the ''x'' and ''y'' coordinates are the same for this angle, the tangent ratio is 1.
  
  
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==Resources==
 
==Resources==
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20Functions_%20Unit%20Circle%20Approach/Esparza%201093%20Notes%202.2.pdf Trigonometric Functions: Unit Circle Approach]. Written notes created by Professor Esparza, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20Functions_%20Unit%20Circle%20Approach/Esparza%201093%20Notes%202.2.pdf Trigonometric Functions: Unit Circle Approach]. Written notes created by Professor Esparza, UTSA.
 +
* [https://en.wikibooks.org/wiki/High_School_Trigonometry/Defining_Trigonometric_Functions Defining Trigonometric Functions], WikiBooks: High School Trigonometry
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* [https://en.wikibooks.org/wiki/Trigonometry/The_Unit_Circle The Unit Circle], WikiBooks: Trigonometry
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 +
==Licensing==
 +
Content obtained and/or adapted from:
 +
* [https://en.wikibooks.org/wiki/Trigonometry/The_Unit_Circle The Unit Circle, WikiBooks: Trigonometry] under a CC BY-SA license
 +
* [https://en.wikibooks.org/wiki/High_School_Trigonometry/Defining_Trigonometric_Functions Defining Trigonometric Functions, WikiBooks: High School Trigonometry] under a CC BY-SA license

Latest revision as of 12:53, 25 October 2021

The Unit Circle

Consider an angle in standard position, such that the point (x, y) on the terminal side of the angle is a point on a circle with radius 1.

Defining Trigonometric Functions Figure 4.svg

This circle is called the unit circle. With r = 1, we can define the trigonometric functions in the unit circle:

Notice that in the unit circle, the sine and cosine of an angle are the x and y coordinates of the point on the terminal side of the angle. Now we can find the values of the trigonometric functions of any angle of rotation, even the quadrantal angles, which are not angles in triangles.


The center of the unit circle is the origin '"`UNIQ--postMath-00000007-QINU`"', point '"`UNIQ--postMath-00000008-QINU`"'. Creating line segments starting from point '"`UNIQ--postMath-00000009-QINU`"' to '"`UNIQ--postMath-0000000A-QINU`"' on the '"`UNIQ--postMath-0000000B-QINU`"'-axis will create two line segments. From this, we learn that, in terms of '"`UNIQ--postMath-0000000C-QINU`"', point '"`UNIQ--postMath-0000000D-QINU`"'and other properties.


The center of the unit circle is the origin , point . Creating line segments starting from point to on the -axis will create two line segments. From this, we learn that, in terms of , point and other properties.


Labeled unit circle


We can use the figure above to determine values of the trig functions for the quadrantal angles. For example, . .

Example 1

Use the unit circle above to find each value:

a. cos 90°

b. cot 180°

c. sec 0°

Solution:

a. cos 90° = 0

The ordered pair for this angle is (0, 1). The cosine value is the x coordinate, 0.

b. cot 180° is undefined

The ordered pair for this angle is (-1, 0). The ratio is , which is undefined.

c. sec 0° = 1

The ordered pair for this angle is (1, 0). The ratio is is = 1.

There are several important angles in the unit circle that you will work with extensively in your study of trigonometry: 30°, 45°, and 60°. To find the values of the trigonometric functions of these angles, we need to know the ordered pairs. Let's begin with 30°.

The terminal side of the angle intersects the unit circle at the point . (You will prove this in one of the review exercises.). Therefore we can find the values of any of the trig functions of 30°. For example, the cosine value is the x-coordinate, so cos (30°) = . Because the coordinates are fractions, we have to do a bit more work in order to find the tangent value:

In the review exercises you will find the values of the remaining four trig functions of this angle. The table below summarizes the ordered pairs for 30°, 45°, and 60° on the unit circle.

Table 1.10
Angle x coordinate y coordinate
30°
45°
60°

We can use these values to find the values of any of the six trig functions of these angles.

Example 2

Find the value of each function.

a. cos (45°)

b. sin (60°)

c. tan (45°)

Solution:

a. cos (45°) =

The cosine value is the x coordinate of the point.

b. sin (60°) =

The sine value is the y coordinate of the point.

c. tan (45°) = 1

The tangent value is the ratio of the y coordinate to the x coordinate. Because the x and y coordinates are the same for this angle, the tangent ratio is 1.


Resources

Licensing

Content obtained and/or adapted from:

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