# Unit Circle Trig Functions Chart

**Unit Circle Trig Functions Chart** – The unit circle can be used to define the right triangle relationships known as sine, cosine, and tangent. © HowStuffWorks 2021

You probably have an intuitive idea of what a circle is: the shape of a basketball hoop, a wheel, or a quarter. You may even remember from high school that a radius is any straight line that starts at the center of a circle and ends at its perimeter.

## Unit Circle Trig Functions Chart

A unit circle is just a circle that has a radius of length 1. But it often comes with some other bells and whistles.

### Unit 8: Trigonometry

The unit circle can be used to define the right triangle relationships known as sine, cosine, and tangent. These relationships describe how the angles and sides of a right triangle are related to each other. Say, for example, we have a right triangle with an angle of 30 degrees, and whose longest side, or hypotenuse, is 7. We can use our predefined right triangle relationships to figure out the length of the two remaining sides of the triangle. .

This branch of mathematics, known as trigonometry, has practical everyday applications such as construction, GPS, plumbing, computer games, engineering, carpentry and flight navigation.

To help us out, we’ll remind you of a trip to Unit Pizza Palace. Take a few moments to memorize the following until you can recite without looking:

### File:unit Circle Angles Color.svg

Imagine one whole pizza, cut into four equal slices. In mathematics, we call these four parts of a circle quadrants.

Fig. 2. Unit circle with added quadrant. Quadrant 1 is on the right, quadrant 2 is on the left, quadrant 3 is on the left and quadrant 4 is on the right.

We can use coordinates (x, y) to describe any point along the outer edge of the circle. The x coordinate represents the distance traveled to the left or right of the center. The y coordinate represents the distance traveled up or down. The x-coordinate is the cosine of the angle formed by the point, the origin, and the x-axis. The y coordinate is the sine of the angle.

## Basics Of Trigonometry: Definition, Table, Examples

In the unit circle, a straight line traveling straight through the center of the circle will meet the edge of the circle at the coordinate (1, 0). If we instead went up, left, or down, we would touch the perimeter at (0, 1), (-1, 0), and (0, -1), respectively.

The four related angles (in radians, not degrees) all have a denominator of 2. (A radian is the angle made when you take a radius and wrap it around a circle. A degree measures angles by distance traveled. A circle is 360 degrees or 2π radians).

Counters start at 0, start at coordinates (1, 0) and count counter-clockwise by 1π. This process will yield 0π/2, 1π/2, 2π/2, and 3π/2. Simplify these fractions to get 0, π/2, π, and 3π/2.quad

## Solving Trig Equations

Start with “3 pies.” Look at the y-axis. The radian angles directly to the right and left of the y-axis have a denominator of 3. Each remaining angle has a numerator that includes the mathematical value pi, written as π.

“3 pies for 6” is used to recall the remaining 12 angles in the standard unit circle, with three angles in each quadrant. Each of these angles is written as a fraction.

“for $6” should remind us that in each quadrant the remaining denominators are 4 and then 6.

#### Unit Circle Labeled With Quadrantal Angles And Values

In quadrant 2 (upper left quarter of the circle) put 2, then 3, then 5 in front of π.

Your first angle in quadrant 2 will be 2π/3. Adding 2 in the numerator and 3 in the denominator will give 5. Look at the right angle across in quadrant 4 (lower right quarter of the circle). Put this 5 in the numerator before the π. Repeat this process for the other two corners in quadrants 2 and 4.

We will repeat the same procedure for quadrants 1 (upper right) and 3 (lower left). Remember, just as x is the same as 1x, π is the same as 1π. So we add 1 to all the denominators in quadrant 1.

### Trig: Unit Circle

The process of specifying angles in degrees (instead of radians) is described at the end of this article.

The “2” in “2 square tables” should remind us that the remaining 12 pairs of coordinates have a denominator of 2.

“Square” reminds us that the numerator of each coordinate includes the square root. We only start with quadrant 1 to keep things simple. (Hint: Remember that the square root of 1 is 1, so these fractions can be simplified to just 1/2.)

### Unit Circles And Standard Position (video & Practice)

“1, 2, 3” shows us the sequence of numbers under each square root. For the x coordinates of quadrant 1, we count from 1 to 3, starting from the top coordinate and going down.

Y coordinates have the same counters, but count from 1 to 3 in the opposite direction, from bottom to top.

Quadrant 3 reverses the x and y coordinates from quadrant 1. The x and y coordinates are also negative.

#### Trigonometry Table: Formulas, Meaning, Examples

Like quadrant 3, quadrant 4 also inverts the x and y coordinates from quadrant 1. But only the y coordinates are negative.

You may want to specify angles in degrees instead of radians. To do this, start at 0 degrees at coordinates (1, 0). From there we add 30, 15, 15 and then 30. In quadrant 1 we add 30 to 0 to get 30, add 15 to 30 to get 45, add 15 to 45 to get 60 and add 30 to 60 to get 90.

Then we repeat the process for the remaining quadrants, adding 30, 15, 15 and 30 until we reach the end of the circle. Thus, quadrant 4 will have angles ranging from 270 to 330 degrees (see Figure 10).

### Resourceaholic: Teaching Trigonometry

Earlier in the article, we mentioned that the unit circle can be used to find two unknown sides of a right triangle at an angle of 30 degrees, and that its longest side, or hypotenuse, has a length of 7. Let’s try.

Note where 30° is on the unit circle. Use that line and the x-axis to make a triangle as follows.

Fig. 10. Use the unit circle to find the two unknown sides of a 30 degree right triangle

### The Unit Circle And Trigonometric Ratios For Any Angle

In the unit circle, every line that starts from the center of the circle and ends at its perimeter has length 1. Therefore, the longest side of this triangle will have length 1. The longest side of a right triangle is also known as the “hypotenuse”. The point where the hypotenuse touches the perimeter of the circle is at √3/2, 1/2.

So, we know that the base of the triangle (on the x-axis) has a length of √3/2, and the height of the triangle is 1/2.

Another way to think about it is that the base is √3/2 times the length of the hypotenuse and the height is 1/2 times the length of the hypotenuse.

#### Solved Fill In The Blanks And The Complete The Unit Circle:

So if instead, the hypotenuse is 7, the base of our triangle will be 7 x √3/2 = 7√3/2. The height of the triangle will have a length of 7 x 1/2 = 7/2.

Special offer on antivirus software from HowStuffWorks and TotalAV Security Try our crossword puzzle! Can you solve this puzzle? The unit circle is the golden key to really understanding trigonometry. Like many ideas in mathematics, its simplicity makes it beautiful.

Sin, cos and fire measurements become clearer when you see them on the chart. Take a moment to soak in what they mean.

### Discovering The Trig Functions On The Unit Circle

The unit circle clearly shows the trig function because the radius is 1. The hypotenuse does not change the value of sin, cos, and tan.

A little time spent memorizing them now will save you a LOT of time and work in the future.

Memorization sounds like a pain, but don’t worry, there are some tricks to help. Let’s start with the values for sin.

## Unit Circle (video)

This can be easily remembered by thinking of the stop sign, and cos felt. See where this is going?

Tan also leads to a nice pattern, although it doesn’t include 0° and 90° like sin and cos do.

Put all this together and you get a table of special trigonometric values, or a table of the unit circle:

### Trigonometry Equation, Unit Circle Trigonometry Sine Trigonometric Functions, Pi Math, Angle, Text Png

You might like this better, because you can understand it yourself, even in the middle of an exam!

Then use SOH CAH TOA on the triangle. Remember that each interior angle of an equilateral triangle is 60°, so the bisected angle is 30°.

The value of sin, cos and tan remains the same in each quadrant, but the sign changes depending on which quadrant the angle is in.

## Old Fashioned Trig Table

And put them all together. This leads to this very handy chart. Click/tap the image to open the printable PDF.

There it is! Values involving pi, π are called radians. They have a special relationship with circles and are the next step on the path to mastering the unit circle. Calculate the coordinates for a point on the unit circle given the central angle in radians or degrees. You will also get sine, cosine and tangent in the results.

A unit circle is a circle with a radius of 1 u

### Unit Circle Calculator

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