The Unit Circle
The Unit Circle
The unit circle is a circle whose center is the origin and whose radius is 1. It is defined by the equation x2 + y2 = 1.
The most useful and interesting property of the unit circle is that the coordinates of a given point on the circle can be found using only the knowledge of the measure of the angle.
Any radius of the unit circle is the hypotenuse of a right triangle that has a (horizontal) leg of length cos and a (vertical) leg of length sin . The angle is defined as the radius measured in standard position. These relationships are easy to see using the trigonometric functions:
As you can see, because the radius of the unit circle is 1, the trigonometric functions sine and cosine are simplified: sin = y and cos = x. This means that another way to write the coordinates of a point (x, y) on the unit circle is (cos , sin ), where is the measure of the angle in standard position whose terminal side contains the point.
Here’s an example of a typical question on the Math IIC that tests this principle:
What are the coordinates of the point P pictured below?
Point P is the endpoint of a radius of the unit circle that forms a 30º angle with the negative x-axis. This means that an angle of 210º in standard position would terminate in the same position. So, the coordinates of the point are (cos 210º, sin 210º) = (–/2, –1/2). Both coordinates must be negative, since the point is in the third quadrant.
Range
The unit circle also provides a lot of information about the range of trigonometric functions and the values of the functions at certain angles.
For example, because the unit circle has a radius of one and has points all of the form (cos , sin ), we know that:
Tangent ranges from –∞ to ∞, but is undefined at every angle whose cosine is 0. Can you guess why? Look at the formula of tan = sin / cos . If cos = 0, then division by 0 occurs, and the quotient, tan , is undefined.
Radians and Degrees
Radians are another way to measure angles. Sometimes radians will be used in questions, and other times you may choose to use them since they can sometimes be more convenient to use than degrees. Radians will never appear on the Math IC test, but for the Math IIC, it is necessary to know how to convert between radians and degrees.
A degree is equal to 1 /360 of a circle, while a radian is equal to the angle that intercepts an arc of the same length as the radius of the circle. In the figure below, arc AB has length r, and the central angle measures one radian.
When converting between the two measurement systems, use the proportion:
which can be simplified down to:
To convert from degrees to radians: multiply the degree measure by π/ 180. For example, 60º is equal to 60π/ 180 = π/3 radians.
To convert from radians to degrees: multiply the degree measure by 180/ π. For example, π/4 is equal to 180π/ = 45º.
Here are the most important angle measures in degrees and radians:
On the Math IIC, it is sometimes a better idea to work solely in radians rather than convert back and forth between radians and degrees. Using radians is especially easy on graphing calculators that allow you to switch into radian mode.
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