The "Unit Circle" is a circle with a radius of 1 and a center at (0, 0). Because the radius is 1, we can directly measure sine, cosine and tangent. The angle (in radians) that t intercepts forms an arc of length s.Using the formula s=rt, and knowing that r=1, we see that for a unit circle, s=t.
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 t, we can label the intersection of the terminal side and the unit circle as by its coordinates, (x,y). The coordinates x and y will be the outputs of the trigonometric functions f(t) = cost and f(t) = sint, respectively. Thismeans x= cost and y= sint.
At t=π/4, which is 45 degrees, the radius of the unit circle bisects the first quadrantal angle. This means the radius lies along the line y=x. A unit circle has a radius equal to 1. So, the right triangle formed below the line y=x has sides x and y(y=x), and a radius = 1. [tex]x^2+y^2=1, \\ y=x[/tex], then [tex]y=x= \frac{ \sqrt{2} }{2} [/tex] (because 45° is in first quadrant).
The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle. Because the y-value is equal to the sine of t, and the x-value is equal to the cosine of t, the tangent of angle t can also be defined as sint/cost, cost≠0.
For example, tan 45°=1.
