# 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 . 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 , and . 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.

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).

Since we are asked for the distance, there is no negative distance thus the only answer is

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 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 and the side opposite the 60-degree angle is .

*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, and .

*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 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 . Similarly, the horizontal segment of the triangle opposite the other 45-degree angle or .

**Coordinates of Special angle in Unit circle**

Remember:

When predicting the coordinate of one angle, and

, and , and 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 . Since is married to the value of

### Dan

#### Latest posts by Dan (see all)

- 2016 MMC Schedule - November 4, 2015
- 2014 MTAP reviewer for Grade 3 - September 30, 2015
- 2015 MTAP reviewer for 4th year solution part 1 - August 22, 2015

Hi there would you mind letting me know which web host you’re using? I’ve loaded your blog in 3 completely different web browsers and I must say this blog loads a lot quicker then most. Can you suggest a good hosting provider at a fair price? Thanks, I appreciate it!

I’ve been exploring for a little for any high-quality articles or blog posts on this sort of area . Exploring in Yahoo I at last stumbled upon this site. Reading this info So i’m happy to convey that I’ve an incredibly good uncanny feeling I discovered exactly what I needed. I most certainly will make certain to do not forget this web site and give it a glance regularly.

hmipXi Spot on with this write-up, I really feel this website needs a lot more attention. I all probably be back again to see more, thanks for the information!

664208 182825I believe 1 of your commercials caused my internet browser to resize, you could properly want to put that on your blacklist. 102773

This post is on 18 spot in google’s search results, if you want more visitors, you should

build more backlinks to your website, there is one trick to get free, hidden backlinks from authority forums, search in google:

how to get hidden backlinks from forums

Wow! Thank you! I continually wanted to write on my site something like that. Can I implement a portion of your post to my website?

Hello Dear, are you actually visiting this site regularly, if

so afterward you will without doubt obtain nice knowledge.

Major thankies for the article post.Thanks Again. Will read on…

Hello! Would you mind if I share your blog with my facebook group?

There’s a lot of folks that I think would really appreciate your content.

Please let me know. Thanks

I’m not sure where you are getting your info, but great

topic. I needs to spend some time learning more or understanding more.

Thanks for fantastic info I was looking for this info

for my mission.

Hello! Quick question that’s totally off topic. Do you know

how to make your site mobile friendly? My site looks weird when browsing

from my iphone. I’m trying to find a theme or plugin that might be able

to correct this issue. If you have any recommendations, please share.

Cheers!

My brother recommended I might like this blog. He was once

totally right. This put up actually made my day. You cann’t

consider simply how so much time I had spent

for this information! Thanks!

If you would like to grow your familiarity only keep visiting this website and be updated with the

hottest news update posted here.

Appreciate this post. Let me try it out.

I’m not sure why but this blog is loading incredibly slow for me.

Is anyone else having this problem or is it

a issue on my end? I’ll check back later and see if the problem still exists.

Hi, this weekend is pleasant in support of me, for the

reason that this moment i am reading this enormous informative piece of writing here at my home.

Thanks for sharing, this is a fantastic blog article.Thanks Again. Will read on…

I’m amazed, I have to admit. Rarely do I encounter a blog that’s both educative and engaging, and let me tell you, you have hit the nail on the head.

The problem is something which too few men and women are

speaking intelligently about. I am very happy I

came across this in my search for something regarding

this.

I am really enjoying the theme/design of

your blog. Do you ever run into any browser compatibility

issues? A small number of my blog readers have complained about

my website not working correctly in Explorer but looks great in Safari.

Do you have any suggestions to help fix this problem?

402594 738289among the very best system I know, thank you quite considerably . 922787

What’s up mates, how is everything, and what you want to say

about this post, in my view its genuinely amazing in support

of me.

Link exchange is nothing else except it is simply placing the other person’s website link on your page at suitable place and other

person will also do similar in support of you.