The Unit Circle and Trigonometric Functions

The Unit Circle and Trigonometric Functions

A unit circle is basically a circle centered at the origin that has a radius of 1. With a unit circle, we can apply trigonometric functions to any angle. The unit circle is generally determined in the Cartesian coordinate plane. The interior of a unit circle is known as the open unit disk, and the interior of a unit circle with the combination of the unit circle itself is known as the closed unit disk. It is usually denoted with the symbol S1. The formula to a unit circle is x2+y2=1.

The unit circle provides a visual way of representing and thinking about trigonometric functions. And since it has a radius of 1, it has a special relevance regarding trigonometric functions like sine and cosine. If, in a standard position, the point on the circle is towards the terminal side of an angle, then the sine of that angle will be the y-coordinate, and on the other side, x-coordinate will be the cosine of such an angle.

Trigonometric functions on the unit circle 

A unit circle is centered at the origin, and since it has a radius of 1, it becomes easy to measure cos, sin, and tan of trigonometric functions. Now, as the unit circle is centered in the origin, at any point, it has coordinates of x and y. These x and y coordinates given by an angle at a point on the unit circle are defined by the trigonometric functions. To be precise, the x and y coordinates of a unit circle are equal to the cos theta and the sin theta of trigonometric functions. Here, we can measure cos theta =x, and sin theta=y as any values of theta made by a radius line with the x-axis (positive), cos theta and sin theta is represented by the coordinates of the endpoint of that radius. 

Now, we have cos and sin, and in order to find tan, the application of the formula of tan theta= sin theta/ cos theta will provide the right answer. Or it can also be put as tan=y/x. 

The unit circle is divided into four quarters by the x and y-axis that gives a summation of 360 degrees and is called quadrants. It is to note that the lines of the circle are 90 degrees, 180 degrees, 270 degrees, and 360 degrees angles, and the cos theta at 90 degrees and 270 degrees is equal to 0. Here, the tan values will be indeterminate.  

With right angles, trigonometric functions can be applied to angles that measure only to a limited degree of 90 degrees. But with the unit circle, we can apply trigonometric functions to angles of more than 90 degrees.

Some considerations of the unit circle:

In the unit circle, with the use of x=cos and y=sin, we can measure x- and y-

The unit circle shows that the trigonometric functions are periodical as they provide an outcome in a repeated set of values with regular intervals.

Unit circle and trigonometric identities

We learned that the unit circle provides the trigonometric function or identities of cos, tan, sin. These unit circle identities can also be applied to measure other trigonometric identities such as cotangent, cosecant, secant. While the identities such as cotangent, secant, and cosecant are reciprocals of tangent, cosine, and sine, tan theta can be achieved by dividing the value of sin and cos theta. Also, cot theta can be obtained by diving cos and sine theta.

So we learned that the unit circle is a circle with a radius of 1 centered at the origin. It has special relevance to trigonometric functions because of its capability of calculating trigonometric identities. 

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