Making Sense of the Unit Circle
Find Cos, Sin and Tan
Degrees to Radians Conversion Table
- Conversion Table
View the Unit Circle Deck

Find Cos, Sin and Tan

Three out of five values, such as $\frac{1}{2},\frac{\sqrt2}{2},\frac{\sqrt3}{3}$,used in the coordinates of the Unit Circle, can be found in Special Right Triangles. There are two types of Special Right Triangles:

$\mathbf{30}°-60°-90°$ triangle is a right triangle with acute angles of 30° and 60°.

Let triangle ABC be $30°-60°-90°$ triangle with the hypotenuse, that is 1 unit long, such as AB=1.

Right triangle with acute angle of 30° has an incredibly unique property: the leg that is opposite to a 30° angle in a right triangle, is equal to a half of its hypotenuse. Therefore, we establish that the leg BC=$BC=\frac{1}{2}$.

Recall that if two sides of a right triangle are given, then we can find its third side, using Pythagorean Theorem, to get AC=$\frac{\sqrt3}{2}$

$\mathbf{45}°-45°-90°$ triangle is a right isosceles triangle with congruent acute angles that both have the measure of 45°.

Let triangle ABC be 45°-45°-90° triangle with the hypotenuse that is 1 unit long, such as AB=1.

Recall that the legs of an isosceles triangle are congruent and equal to $\frac{\sqrt2}{2}$ units.

Just like we defined the relationship between the sides of a triangle, using the Pythagorean Theorem, we can define the relationship between the sides and angle in a right triangle. To do so, we must introduce trigonometric functions Sine, Cosine, and Tangent of a certain acute angle $\theta$, as ratios:

\[\sin{\theta}=\frac{length\ of\ leg\ opposite\ angle\ \theta}{Length\ of\ hypotenuse}\]

\[\cos{\theta}=\frac{length\ of\ leg\ adjacent\ angle\ \theta}{Length\ of\ hypotenuse}\]

\[\tan{\theta}=\frac{length\ of\ leg\ opposite\ angle\ \theta}{length\ of\ leg\ adjacent\ angle\ \theta}\]

Let us, then, apply these formulas to the Special Right Triangles with the hypotenuse of 1 unit long. While the values of the Sine and Cosine appear to be the length of the sides of these triangles, the values of the Tangent are simply the ratio of the legs.

Using 30°-60°-90° triangle, we define the values of trigonometric functions of 30° and 60°:

\[sin30°= \frac{\sqrt1}{2}, cos30°= \frac{\sqrt3}{2}, tan30°= \frac{\sqrt3}{3},\]

\[sin60°= \frac{\sqrt3}{2}, cos60°= \frac{\sqrt1}{2}, tan60°= {\sqrt3}\]

Note that, when we work with 30° and 60° angles, our possible Sine and Cosine values are either $\frac{1}{2}$ or $\frac{\sqrt3}{2}$. It is because we determine the values of Sine and Cosine using the same exact triangle. That is why, the value of Sine of 30° is the same as the value of Cosine of 60°, and vice versa:

\[sin30°=cos60°= \frac{\sqrt1}{2}\]

\[sin60°=cos30°= \frac{\sqrt3}{2}\]

Now, let us express trigonometric values of 45° angle, using 45°-45°-90° triangle:

\[sin45°= \frac{\sqrt2}{2}, cos45°= \frac{\sqrt2}{2}, tan45°= 1\]When we work with the 45° angles, both, Sine and Cosine values, are the same and equal to $\frac{\sqrt2}{2}$. The angle of 45° is the only acute angle, the Sine and Cosine of which have the same value. As a result, the value of the Tangent of this angle is equal to 1.

Previously we introduced the Unit Circle and Special Right Triangles. Now let us put them all together to define the Sine, Cosine, and Tangent on the Unit Circle:

Terminology

Let us define the generalized angle. An angle in a coordinate plane is called generalized angle, when its vertex is at the origin and one of its side, called the initial side, is fixed at the positive direction of the x-axis. The other side of the angle is called the terminal side.

This way we can form new angles by simply moving the terminal side of an angle clockwise or counterclockwise. If the terminal side moves counterclockwise, then the angle is positive, otherwise, the angle is negative.

Let us introduce the generalized angle $\theta$ with the terminal side intersecting the unit circle at point $P\left(x,y\right)$ and define the values of trigonometric function, using the Special Right Triangles on the Unit Circle.

The sign of trigonometric function carries an important information and may be associated with the x- and y-axis. That is why we will go over each function individually.

Definition of $\mathbf{sin} {{\theta}}$

The function, that to any generalized angle $\theta$ assigns the ordinate (the y-coordinate) of the point $P\left(x,y\right)$ from the Unit Circle, is called the Sine of $\theta$:

\[\sin{\theta}=y\]

The Sine of angles with the terminal side in the first and second quadrants take on positive values, because the y-coordinate of every point in these quadrants is positive. Thus, all angle with terminal side in the third and fourth quadrants will take on negative values of the Sine.

Definition of $\mathbf{cos} {{\theta}}$

The function, that to any generalized angle $\theta$ assigns the abscissa (the x-coordinate) of the point $P\left(x,y\right)$ from the Unit Circle, is called the Cosine of $\theta$:
$\cos{\theta}=x$

The Cosine of angles with terminal side in the first and fourth quadrants take on positive values, because the x-coordinate of every point in these quadrants is positive. Thus, all angle with terminal side in the second and third quadrants will take on negative values of the Cosine.

Definition of $\mathbf{tan} {{\theta}}$

The function, that to any generalized angle \theta assigns the ratio of abscissa to ordinate (the ratio of y-coodrinate to x-coordinate) of the point $P\left(x,y\right)$ from the Unit Circle, is called the Tangent of $\mathbf{\theta}$:

\[\tan{\theta}=\frac{y}{x}\]

The Tangent of angles, which values of Sine and Cosine have the same sign, take on positive values. That is why the values of the Tangent in the first and third quadrants are always positive, while they take on negative values in the quadrants two and four.

Main Concept

When point $P\left(x,y\right)$ belongs to the first quadrant, we simply apply the definition of trigonometric functions on the Unit Circle.

Knowing the sides of the Special Right Triangle, we may then determine the coordinates of the point P: the length of horizontal leg of this triangle is equal to the value of x-coordinate, and the length of vertical leg of this triangle is equal to the values of y-coordinate.

It is important to understand by now that the values of trigonometric functions may vary in sign in the second, third, and fourth quadrants. The sign depends on (1) the quadrant and (2) trigonometric function in this quadrant.

Prior to go beyond the main concept, we should see its application in the first quadrant.

Example 1

Let angle $\theta$ be equal to 30°. The value of trigonometric functions can be found by simply drawing the unit circle, angle $\theta$, and a triangle on a coordinate plane.

Angle $\theta$ intersects the Unit Circle at point $P\left(x,y\right)$.

When the triangle is constructed and the given angle is shown on a diagram, we notice that it is 30°-60°-90° triangle. We already know the length of its side, so let also show them on a diagram:

Vertical side of a triangle, that equals to $\frac{1}{2}$ , corresponds to the y-coordinate, while horizontal side of a triangle, that equals to $\frac{\sqrt3}{2}$ , corresponds to the x-coordinate:
x=$\frac{\sqrt3}{2} and y=\frac{1}{2}$.

Recall the formulas:

\[\sin{\theta}=y,\ \ \cos{\theta}=x,\ \ \tan{\theta}=\frac{y}{x}\]

Substitute the values of x and y to get:

\[sin 30° =1/2, cos⁡ 30°$ =√3/2, tan⁡ 30°$ =√3/3\]

Essentially, this is how we determine the values of trigonometric functions of any acute angle.

Reference Angle

So far, we have defined trigonometric functions of $30°$, $60°$ and $45°$ angles, where point P\left(x,y\right) was in the first quadrant. Let see what happens when this point is located in the other quadrants. Another way to think about it is our generalized angle is greater than $90°$.

No matter where point P is located, we always construct a right triangle at x-axis, so one of its legs is perpendicular to x-axis.

When the triangle is constructed, angle $\theta$ appears outside of this triangle. That is why, we find the measure of angle $\alpha$. Practically, we find the angles of this triangle to determine its sides.

Recall that the value of angle $\alpha$ can be determined using the Angle Measure in the Unit Circle:

In the second quadrant, we subtract the given angle from 180°: $\alpha=180°-θ$.
In the third quadrant, we subtract 180° from the given angle: $\alpha=\theta-180°$.
In the fourth quadrant, we subtract the given angle from 360°: $α=360°-θ$.

This angle we call a reference angle. The reference angle is a positive acute angle, formed by an angle’s terminal side and the x-axis.

Note, that a generalized angle with its terminal side in the first quadrant is equal to its reference angle.

Beyond Main Concept

We have just discussed the last piece of information, a reference angle, needed to make sense of the Unit Circle. To enhance our experience and efficiency when working with trigonometric function, we should classify the angles on the Unit Circle. We use “the approach” as the main factor of our classification.

Thus, all generalized angles can be grouped into three different classes:

Acute angles, such as $30°$, $45°$, and $60°$.
Recall that with these angles, we simply apply the Main Concept to determine the values of trigonometric functions from the Unit Circle. We already described this approach earlier in this article, see the Main Concept and the Example 1.
Angles on the x- and y-axis, such as $0°, 90°, 180, 270°$.
It is quite easy to find the values of Sine and Cosine of these angles because they are the coordinates of intersection the Unit Circle with x- and y-axis. Recall that the radius of the circle is 1 unit long. That is why our values can only be 0 or 1 in this case.

Let us draw the unit circle and annotate the angles of $0°, 90°, 180, 270°$ and the coordinates of the intersection points:

Recall that the value of Cosine is x-coordinate and the value of Sine is y-coordinate:

cos⁡ $0°=1$ and sin⁡ $0°=0$ are presented by the point $\left(1,0\right)$ at $\theta=0°$

cos⁡ $90°=0$ and sin⁡$90°=1$ are presented by the point $\left(0,1\right)$ at $\theta=90°$

cos⁡ $180°=-1$ and sin⁡ $180°=0$ are presented by the point $\left(-1,0\right)$ at $\theta=180°$

cos⁡ $270°=0$ and sin⁡ $270°=-1$ are presented by the point $\left(0,-1\right)$ at $\theta=270°$

While the Sine and Cosine can be spotted on the Unit Circle, the Tangent, on another hand, appears as a ratio of the Sine to Cosine, but why is that?

Recall that the $\sin{\theta}$ corresponds to y and the $\cos{\theta}$ corresponds to x, while the $\tan{\theta}$ is the ration of y to x:

\[\sin{\theta}=y,\ \ \cos{\theta}=x,\ \ \tan{\theta}=\frac{y}{x}\]

That is why, we can express the Tangent in terms of the Sine and Cosine, by replacing x and y values with the $\cos{\theta}$ and $\sin{\theta}$ respectively:

\[\tan{\theta=\frac{\sin{\theta}}{\cos{\theta}}}\]

Recall from algebra that denominator of any fraction, the bottom of the fraction, can never be equal to 0. That is why, the values of Tangent, for which the value of Cosine is equal to 0, cannot be determine:

\[tan⁡ 90°=sin⁡ 90° /cos⁡ 90°\]

\[tan⁡ 90°=1/0\]

\[tan⁡ 90° is \ \ \ undefined\]

\[tan⁡ 270°=sin⁡ 270° /cos⁡ 270°\]

\[tan⁡ 270°=(-1)/0\]

\[tan⁡ 270° is \ \ \ undefined\]

Non-acute angles, such as $120°, 135°, 150°, 210°$, and the rest of the angles, that are shown on the Unit Circle

Working with these angles, we must go beyond the Main Concept. Practically, we apply the Main Concept to the reference angle $\alpha$, instead of the given angle $\theta$.

That is why, to determine the values of trigonometric functions, we must perform the following steps:

First, we should find a reference angle.

Based on the way a generalized angle and reference angle are defined, we can establish that the values of trigonometric function of these angles are the same or opposite to each other:

\[\sin{\theta}=\pm\sin{\alpha}\]

\[\cos{\theta}=\pm\cos{\alpha}\]

\[\tan{\theta}=\pm\tan{\alpha}\]

Then, we must decide on what sign, plus (+) or minus (–), should be kept in the equations above. Recall that it is based on two things: (1) the quadrant, where the terminal side of angle \theta is located, and (2) the sign of a trigonometric function in this quadrant.
Finally, considering that $\alpha$ is a reference angle of 30°, 45°, or 60°, we can simply determine the values of trigonometric function of the angle $\alpha$ from the Unit Circle. Recall that we always work with one of the Special Right Triangles.

Now, let us show how to apply this algorithm.

Example 2

Let us find the values of the Sine, Cosine, and Tangent of 120° angle. We begin with drawing the Unit Circle, angle $\theta=120°$ , and a triangle on a coordinate plane. In this case, point P is in the second quadrant:

First, step 1, we find the reference angle, by subtracting the given angle from 180°:

\[\alpha=180°-θ\]

\[\alpha=180°-120°\]

\[\alpha=60°\]

Knowing the reference angle, we can establish the following:

\[sin⁡ 120° =±sin 60°\]

\[cos⁡ 120° =±cos 60°\]

\[tan⁡ 120° =±tan 60°\]

Then, step 2, we determine what sign we must keep in this statements.

Based on the measure of the given angle, $\theta=120°$, (1) its terminal side is in the second quadrant, and (2) the Sine value is positive in this quadrant, while the Cosine and Tangent values are negative.

That is how we can remove unnecessary signs in the statements from step 1 to get:

\[sin⁡ 120° = sin⁡ 0°\]

\[cos⁡ 120° =-cos 60\]

\[tan⁡ 120° =-tan 60\]

Finally, we can apply the concept of 30-60-90 triangle. It states that $sin⁡ 60°=√3/2$, $cos⁡ 60°=1/2$, and $tan⁡ 60° =√3$. Substitute these values in the statements from step 2 to get:

\[sin⁡ 120° =sin⁡ 0° = √3/2\]

\[cos⁡ 120° =-cos 60 =-1/2\]

\[tan⁡ 120° =-tan 60 =-√3\]

This is how we define the values of trigonometric functions of non-acute generalized angles.

Recall that the Unit Circle diagram consist of sixteen different angle values:

$0°,30°,45°,60°,90°,120°,135°,150°,180°,210°,225°,240°,270°,300°,315°,330°$ .

Why are there only sixteen of them? What has happened to $360°$ angle?

Well, the answers to these questions are the same: trigonometric functions are periodic functions, meaning that they repeat their values.

The period of the Sine and Cosine functions is $2\pi$ or $360°$. It means that their values repeat itself every $2\pi$ or $360°$.

The period of the Tangent, on the other hand, is actually $\pi$ or $180°$. Similarly to the Sine and Cosine values, the Tangent values repeat itself every $\pi$ or $180°$.

That is why, the values of trigonometric functions of $360°$ are the same as the values of these functions of $0°$:

\[sin 360° =sin⁡ 0°\]

\[cos 360° =cos⁡ 0°\]

\[tan 360° =tan⁡ 0°\]

While these statements are true for all trigonometric functions, there is a similar concept that can only be applied to the Tangent function.

Based on the period of the Tangent, the value of the Tangent of $180°$ is the same as the value of Tangent of $0°$ :

\[tan⁡ 180° =tan⁡ 0°\]

Practically speaking, when the given angle is greater than $360°$ (or $180°$ for the Tangent), we may simply subtract the period, $360°$ (or $180°$ for the Tangent), from the given angle. As a result, such approach makes some problems easier and help us work with trigonometric functions more efficiently:

\[sin⁡ 405° =sin (405°- 360°)=sin⁡ 45°\]

\[cos⁡ 390° =sin (390°- 360°)=cos⁡ 30°\]

\[tan⁡ 240° =tan (240°- 180°)=tan⁡ 60°\]

We can see it closely when all trigonometric values are presented in a table format. Note that the value of Tangent of $60°$ is the same as the value of Tangent of $240°$ :

Functions	cos	sin	tan	sec	csc	cot
0	1	0	0	1	undefined	undefined
30°	$\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}$	$\frac{\mathrm{1}}{\mathrm{2}}$	$\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}$	$\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}$	2	$\sqrt{\mathrm{3}}$
45°	$\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}$	$\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}$	1	$\sqrt{\mathrm{2}}$	$\sqrt{\mathrm{2}}$	1
60°	$\frac{\mathrm{1}}{\mathrm{2}}$	$\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}$	$\sqrt{\mathrm{3}}$	2	$\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}$	$\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}$
90°	0	1	undefined	undefined	1	0
120°	$\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}$	$\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}$	$\mathrm{-}\sqrt{\mathrm{3}}$	-2	$\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}$	$\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}$
135°	$\mathrm{-}\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}$	$\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}$	-1	$\mathrm{-}\sqrt{\mathrm{2}}$	$\sqrt{\mathrm{2}}$	-1
150°	$\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}$	$\frac{\mathrm{1}}{\mathrm{2}}$	$\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}$	$\mathrm{-}\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}$	2	$\mathrm{-}\sqrt{\mathrm{3}}$
180°	-1	0	0	-1	undefined	undefined
210°	$\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}$	$\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}$	$\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}$	$\mathrm{-}\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}$	-2	$\sqrt{\mathrm{3}}$
225°	$\mathrm{-}\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}$	$\mathrm{-}\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}$	1	$\mathrm{-}\sqrt{\mathrm{2}}$	$\mathrm{-}\sqrt{\mathrm{2}}$	1
240°	$\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}$	$\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}$	$\sqrt{\mathrm{3}}$	-2	$\mathrm{-}\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}$	$\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}$
270°	0	-1	undefined	undefined	-1	0
300°	$\frac{\mathrm{1}}{\mathrm{2}}$	$\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}$	$\mathrm{-}\sqrt{\mathrm{3}}$	-2	$\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}$	$\mathrm{-}\sqrt{\mathrm{3}}$
315°	$\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}$	$\mathrm{-}\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}$	-1	$\sqrt{\mathrm{2}}$	$\sqrt{\mathrm{2}}$	-1
330°	$\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}$	$\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}$	$\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}$	$\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}$	2	$\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}$

Summary

It is important to be able to find all values of trigonometric functions of any generalized angle. That is why, one may assume that the entire table above needs to be memorized. Luckily for us, we can use a different approach, the one that involves the Unit Circle diagram:
There are four main concentric circles (circles with the same center but different radii) on the Unit Circle diagram. Each circle is color-coded and has its own title, located at $90°$. Let us outline them all starting with the smallest circle, then work our way to the biggest one:

Two inner circles contain all possible values of generalized angle \theta:
- The purple circle contains the values in degrees
- The yellow circle contains the values in radians.
Two outer circles contain all possible values of trigonometric functions, which position corresponds to a generalized angle from the inner circles.
- The pink circle contains the values of Sine and Cosine functions in a form of Cartesian Coordinates $\left(x,y\right)$. Previously, we referred to it as x- and y-coordinates of point P.
- Recall that in the ordered pair $\left(x,y\right)$, x- coordinate represent the value of Cosine function, while y-coordinate represent the value of Sine function:
\[\left(x,y\right)=(\cos{\theta},\sin{\theta)}\]
- The blue circle contains all possible values of Tangent function.
Despite this diagram consist of the circles with different radii, we can use it to locate the Sine, Cosine, and even Tangent values. One of the important parts of the Unit Circle diagram is the location of the angles and values of trigonometric functions that correspond to these angles.

Functions	cos	sin	tan	sec	csc	cot
0	1	0	0	1	undefined	undefined
30°	\(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\)	\(\frac{\mathrm{1}}{\mathrm{2}}\)	\(\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\)	\(\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}\)	2	\(\sqrt{\mathrm{3}}\)
45°	\(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\)	\(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\)	1	\(\sqrt{\mathrm{2}}\)	\(\sqrt{\mathrm{2}}\)	1
60°	\(\frac{\mathrm{1}}{\mathrm{2}}\)	\(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\)	\(\sqrt{\mathrm{3}}\)	2	\(\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}\)	\(\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\)
90°	0	1	undefined	undefined	1	0
120°	\(\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\)	\(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\)	\(\mathrm{-}\sqrt{\mathrm{3}}\)	-2	\(\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}\)	\(\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\)
135°	\(\mathrm{-}\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\)	\(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\)	-1	\(\mathrm{-}\sqrt{\mathrm{2}}\)	\(\sqrt{\mathrm{2}}\)	-1
150°	\(\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\)	\(\frac{\mathrm{1}}{\mathrm{2}}\)	\(\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\)	\(\mathrm{-}\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}\)	2	\(\mathrm{-}\sqrt{\mathrm{3}}\)
180°	-1	0	0	-1	undefined	undefined
210°	\(\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\)	\(\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\)	\(\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\)	\(\mathrm{-}\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}\)	-2	\(\sqrt{\mathrm{3}}\)
225°	\(\mathrm{-}\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\)	\(\mathrm{-}\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\)	1	\(\mathrm{-}\sqrt{\mathrm{2}}\)	\(\mathrm{-}\sqrt{\mathrm{2}}\)	1
240°	\(\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\)	\(\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\)	\(\sqrt{\mathrm{3}}\)	-2	\(\mathrm{-}\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}\)	\(\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\)
270°	0	-1	undefined	undefined	-1	0
300°	\(\frac{\mathrm{1}}{\mathrm{2}}\)	\(\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\)	\(\mathrm{-}\sqrt{\mathrm{3}}\)	-2	\(\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}\)	\(\mathrm{-}\sqrt{\mathrm{3}}\)
315°	\(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\)	\(\mathrm{-}\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\)	-1	\(\sqrt{\mathrm{2}}\)	\(\sqrt{\mathrm{2}}\)	-1
330°	\(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\)	\(\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\)	\(\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\)	\(\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}\)	2	\(\mathrm{-}\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\)

Find Cos, Sin and Tan

Special Right Triangles

Using right triangle to define cosine, sine, and tangent

Determining values of cosine, sine, and tangent from the unit circle

Terminology

Definition of \(\mathbf{sin} {{\theta}}\)

Definition of \(\mathbf{cos} {{\theta}}\)

Definition of \(\mathbf{tan} {{\theta}}\)