![]() The input to the sine and cosine functions is the rotation from the positive x-axis, and that may be any real number. What are the domains of the sine and cosine functions? That is, what are the smallest and largest numbers that can be inputs of the functions? Because angles smaller than 0 and angles larger than 2 π 2 π can still be graphed on the unit circle and have real values of x, y, x, y, and r, r, there is no lower or upper limit to the angles that can be inputs to the sine and cosine functions. Now that we can find the sine and cosine of an angle, we need to discuss their domains and ranges. Identifying the Domain and Range of Sine and Cosine Functions Because each side of the equilateral triangle A B C A B C is the same length, and we know one side is the radius of the unit circle, all sides must be of length 1.Įvaluate sin ( π 3 ). We know the angles in a triangle sum to 180°, 180°, so the measure of angle C C is also 60°. At point B, B, we draw an angle A B C A B C with measure of 60°. (60°), the radius of the unit circle, 1, serves as the hypotenuse of a 30-60-90 degree right triangle, B A D, B A D, as shown in Figure 13. The ( x, y ) ( x, y ) coordinates for the point on a circle of radius 1 1 at an angle of 30° 30° are ( 3 2, 1 2 ). ![]() ![]() cos ( π 6 ) = ± 3 ± 4 = 3 2 Since y is positive, choose the positive root. The cosine function of an angle t 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 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 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 ( x, y ) ( x, y ) coordinates relate to the arc length and angle. The ( x, y ) ( x, y ) coordinates of this point can be described as functions of the angle. Let ( x, y ) ( x, y ) 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 t. Ī unit circle has a center at ( 0, 0 ) ( 0, 0 ) and radius 1 1. This means x = cos t x = cos t and y = sin t. The coordinates x x and y y will be the outputs of the trigonometric functions f ( t ) = cos t f ( t ) = cos t and f ( t ) = sin t, f ( t ) = sin t, respectively. The four quadrants are labeled I, II, III, and IV.įor any angle t, t, we can label the intersection of the terminal side and the unit circle as by its coordinates, ( x, y ). 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 = r t, s = r t, and knowing that r = 1, r = 1, we see that for a unit circle, s = t. The angle (in radians) that t 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. Finding Function Values for the Sine and Cosine Then we can discuss circular motion in terms of the coordinate pairs. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. In this section, we will examine this type of revolving motion around a circle. 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. Looking for a thrill? Then consider a ride on the Singapore Flyer, the world’s tallest Ferris wheel. ![]() Figure 1 The Singapore Flyer is the world’s tallest Ferris wheel. ![]()
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