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Looking for a thrill? Then consider a ride on the Singapore Flyer, the world’s tallest Ferris wheel. Located in Singapore, the Ferris wheel soars to a height of 541 feet—a little more than a tenth of a mile! Described as an observation wheel, riders enjoy spectacular views as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs.
To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in [link] . The angle (in radians) that $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ intercepts forms an arc of length $\text{\hspace{0.17em}}s.\text{\hspace{0.17em}}$ Using the formula $\text{\hspace{0.17em}}s=rt,$ and knowing that $r=1,$ we see that for a unit circle , $\text{\hspace{0.17em}}s=t.\text{\hspace{0.17em}}$
Recall that the x- and y- axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.
For any angle $\text{\hspace{0.17em}}t,$ we can label the intersection of the terminal side and the unit circle as by its coordinates, $\text{\hspace{0.17em}}\left(x,y\right).\text{\hspace{0.17em}}$ The coordinates $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ will be the outputs of the trigonometric functions $\text{\hspace{0.17em}}f(t)=\mathrm{cos}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f(t)=\mathrm{sin}\text{\hspace{0.17em}}t,$ respectively. This means $\text{\hspace{0.17em}}x=\mathrm{cos}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$
A unit circle has a center at $\text{\hspace{0.17em}}(0,0)\text{\hspace{0.17em}}$ and radius $\text{\hspace{0.17em}}1\text{\hspace{0.17em}}$ . In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle $\text{\hspace{0.17em}}1.$
Let $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ be the endpoint on the unit circle of an arc of arc length $\text{\hspace{0.17em}}s.\text{\hspace{0.17em}}$ The $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ coordinates of this point can be described as functions of the angle.
Now that we have our unit circle labeled, we can learn how the $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ coordinates relate to the arc length and angle . The sine function relates a real number $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ to the y -coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the y -value of the endpoint on the unit circle of an arc of length $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ In [link] , the sine is equal to $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the y -coordinate of the corresponding point on the unit circle.
The cosine function of an angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the x -value of the endpoint on the unit circle of an arc of length $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ In [link] , the cosine is equal to $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$
Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: $\mathrm{sin}\text{\hspace{0.17em}}t$ is the same as $\mathrm{sin}(t)$ and $\mathrm{cos}\text{\hspace{0.17em}}t$ is the same as $\mathrm{cos}(t).$ Likewise, $\text{\hspace{0.17em}}{\mathrm{cos}}^{2}t\text{\hspace{0.17em}}$ is a commonly used shorthand notation for ${(\mathrm{cos}(t))}^{2}.\text{\hspace{0.17em}}$ Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer.
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