The cosine function of an angle t equals the x-value of the endpoint on the unit circle of an arc of length t. Its input is the measure of the angle its output is the y-coordinate of the corresponding point on the unit circle. Like all functions, the sine function has an input and an output. More precisely, the sine of an angle t equals the y-value of the endpoint on the unit circle of an arc of length t. The sine function relates a real number t to the y-coordinate of the point where the corresponding angle intercepts the unit circle. Now that we have our unit circle labeled, we can learn how the \left(x,y\right) coordinates relate to the arc length and angle. The \left(x,y\right) coordinates of this point can be described as functions of the angle. Let \left(x,y\right) be the endpoint on the unit circle of an arc of arc length s. In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle 1. ![]() This means x=\cos t and y=\sin t.Ī unit circle has a center at \left(0,0\right) and radius 1. The coordinates x and y will be the outputs of the trigonometric functions f\left(t\right)=\cos t and f\left(t\right)=\sin t, respectively. The four quadrants are labeled I, II, III, and IV.įor any angle t, we can label the intersection of the terminal side and the unit circle as by its coordinates, \left(x,y\right). We label these quadrants to mimic the direction a positive angle would sweep. Recall that the x- and y-axes divide the coordinate plane into four quarters called quadrants. Using the formula s=rt, and knowing that r=1, we see that for a unit circle, s=t. ![]() The angle (in radians) that t intercepts forms an arc of length s. To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in Figure 2. Evaluate sine and cosine values using a calculator.Use reference angles to evaluate trigonometric functions.Identify the domain and range of sine and cosine functions.Find function values for the sine and cosine of the special angles.
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