# Unit Circle

Unit circle is a circle with radius 1 unit. Thus, its name is a unit circle. This simplest circle is the foundation of six  functions in trigonometry namely sine, cosine, tangent, cotangent, cosecant and secant.

In analytical geometry, unit circle has an equation of $x^2+y^2=1$. With center at (0,0) and radius of 1 when graph in Cartesian coordinate plane.

In trigonometry, If we select any point lying in the circumference of the circle and label it (x,y) the value of x is equal to the horizontal distance of the point from the origin. Similarly, the value of y is also the vertical distance from the origin to the point. If we are given with an angle (theta) by which the point terminated, we can say that the value of $x=cos\theta$, and $y=sin\theta$. That is where the two functions born.

If we draw perpendicular line nearest to x-axis as shown in the figure above, we are able to create a right triangle. Pythagoras created a relationship between the legs of the triangle and the length of the hypotenuse. Since the length of the hypotenuse is the radius of the circle which is 1. By Pythagorean Theorem we have this first identity in trigonometry. $x^2+y^2=1^2$ $(cos\theta)^2+(sin\theta)^2=1$ $cos^2\theta +sin^2\theta=1$

Another trigonometric function is born if we take the ratio of sine and cosine. The resulting function is called the tangent.

Three more functions exist as we take the reciprocals of these functions, Secant as reciprocal of cosine, Cosecant as reciprocal of sine, and lastly cotangent as reciprocal of tangent or the ratio of cosine and sine.

These six trigonometric functions are very important in solving right triangle in trigonometry. SOH-CAH-TOA-COH-SHA-CAO is the mnemonic used by our teachers to easily memorize these formulas. Find out how to use them here.

Understanding Unit Circle Mathematically

What is the vertical and horizontal distance of point P from the origin if P terminates 60° above the positive x-axis? The point creates a right triangle inside the unit circle if we draw a perpendicular line with the x-axis and connect it to the given point. Since there is a right angle and the other angle is 60 degrees, for sure the other angle is 30 degrees. Using the property of special triangle, the angle opposite the 30-degree angle is half the hypotenuse.

From the argument above, horizontal distance(x) is ½. Using the Pythagorean Theorem, we can solve for the vertical distance(y). $x^2+y^2=1^2$ $(\displaystyle\frac{1}{2})^2+y^2=1$ $\displaystyle\frac{1}{4}+y^2=1$ $y^2=1-\displaystyle\frac{1}{4}$ $y^2=\displaystyle\frac{3}{4}$ $y=\pm\displaystyle\frac{\sqrt{3}}{2}$

Since we are asked for the distance, there is no negative distance thus the only answer is $\displaystyle\frac{\sqrt{3}}{2}$

Did you know that the value of x and y that we just solve is the same as the value of cos60° and sin60°?

Tricks:

Example: Find cos120° and sin120°

*Draft the unit circle and the angle. *Look closely, close your eyes. Now sleep. No just kidding. Always draw a perpendicular line from the point to the nearest x-axis making a right triangle. *Since we are dealing with special triangles. The segment opposite the 30° is half the length of the radius of the unit circle. The length of the segment opposite the 60° angle is $\sqrt{3}$ times the length of the side opposite the 30-degree angle.

So always start labeling to the point opposite the 30-degree angle. Since the radius of the unit circle is 1. The side opposite the 30-degree angle is $\displaystyle\frac{1}{2}$ and the side opposite the 60-degree angle is $\displaystyle\frac{\sqrt{3}}{2}$.

*The length of the horizontal segment opposite the 30-degree angle is the value of your cos120°.The length of the vertical segment opposite the 60-degree angle is the value of sin120°.

*After obtaining the values, the finishing touch is the sign. *Using the figure above as reference for your answer’s signs, the value of cosine in quadrant 2 is negative and the sign of sine is positive. Thus, $sin120=\displaystyle\frac{\sqrt{3}}{2}$ and $cos120=-\displaystyle\frac{1}{2}$.

*There is one more special triangle, the 45-45-90 also known as the isosceles right triangle.

For example: What is the value of sin45° and cos45°? Again, by dropping perpendicular line to the nearest x-axis, we have a triangle formed which is an isosceles right triangle.

In this triangle, the length of the side opposite the 45-degree angle is $\displaystyle\frac{1}{\sqrt{2}}$ times the side of the hypotenuse. That’s all you need since both angles here are 45 degrees.

Hence, the vertical segment opposite the 45-degree or $sin45=1\cdot\displaystyle\frac{1}{\sqrt{2}}$. Similarly, the horizontal segment of the triangle opposite the other 45-degree angle or $cos45=1\cdot\displaystyle\frac{1}{\sqrt{2}}$.

Coordinates of Special angle in Unit circle Remember:

When predicting the coordinate of one angle, $\displaystyle\frac{\sqrt{3}}{2}$  and $\displaystyle\frac{\sqrt{1}}{2}$, $\displaystyle\frac{\sqrt{2}}{2}$  and $\displaystyle\frac{\sqrt{2}}{2}$, $1$  and $0$  are married to each other.You just need to work on signs. So if you found out that one coordinate using the trick above, you can easily predict the other coordinate.

Example:

What is the coordinate of point P on unit circle if it terminates in 150°? Where is the 30-degree angle facing? Yes! You’re right on the vertical segment.Therefore $sin150=\displaystyle\frac{1}{2}$. Since $\displaystyle \frac{1}{2}$  is married to $\displaystyle\frac{\sqrt{3}}{2}$  the value of $cos150=-\displaystyle\frac{\sqrt{3}}{2}$