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What is cos(0) equal to?The Unit Circle

Unit Circle

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The unit circle

Saturday, March 18th, 2023

The Unit Circle

The unit circle is a circle of radius 1 unit and center at the origin (0,0) of a coordinate plane. Its x-intercepts are (1, 0) and (-1, 0), and the y-intercepts are (0, 1) and (0, -1). To understand the properties of a unit circle, it is helpful to consider a line drawn from the origin to an arbitrary point (x,y) in the circle. This line is referred to as the terminal side, while the angle from the x-axis towards the Terminal side in the counterclockwise direction is known as the Central Angle as shown in the diagram.

A unit circle with terminal side and central angle labelled

The coordinates of the arbitrary point are given by the trigonometric ratios sine and cosine. The x-coordinate is equal to the cosine of the central angle, and the y-coordinate is the sine of the angle. These ratios are defined in terms of the lengths of the sides of a right triangle, just like the Pythagorean theorem.

Example

Example 1. The Sine and Cosine value of

45\degree

Answer:

Consider a 45-degree angle with its terminal side on the unit circle. If we draw a right triangle with this angle, the hypotenuse will have length 1 (since it is a radius of the unit circle), and the other two sides will have identical lengths (since the triangle is a 45-45-90 triangle).

A unit circle with 45 degrees central angle

The Pythagorean theorem tells us that the sum of the squares of the other two sides is equal to the square of the hypotenuse, so by applying the Pythagorean theorem we get:

x^2 + y^2 = 1^2

Since, x = y, we can replace y with x or vice versa:

2x^2 = 1

Solving for x we get:

x = \frac{1}{\sqrt{2}}

hence

y = \frac{1}{\sqrt{2}}

Therefore, the sine and cosine of a 45-degree angle are both equal to

\frac{1}{\sqrt{2}}

Unit Circle in Radians

A radian is the size of the central angle in a unit circle that makes an arc length of 1. An angle of pi radians in a unit circle makes an arc length of pi, which is half the circumference of the unit circle.

Unit Circle Radians of angle 1 rad and pi rad

Radians are a more natural unit of measurement for angles when working with trigonometric ratios and circular functions because they take the relationship between the angle and the arc length of a circle into account.

Below is a unit circle table that indicates standard angles (in radians) and the corresponding sin, cos and tan ratios. [image of unit circle table]

A unit circle table with ratios of standard angles

The Four Quadrants

The unit circle is studied by dividing it into four quadrants along the x- and y-axes as it gives more insight into the behavior of trigonometric ratios. These quadrants are labeled I, II, III, and IV, starting in the upper right and moving counterclockwise.

The trigonometric ratios of angles in a particular quadrant share common characteristics. For example, the sine value of any central angle lying in the 2nd quadrant would be positive, whereas the cosine values in the 2nd quadrant would always be negative. You can derive such characteristics for the other quadrants if you recall that the sine value of the angle is the y-coordinate, and the cosine value is the x-coordinate of a point on the unit circle.

Even sine and cosine values of angles between different quadrants share characteristics if they have the same reference angle. A reference angle is the smallest angle the terminal side makes with the x-axis.

We can generalize this for an angle with a reference angle

\theta

as follows:

Angle	Quadrant	sine	cosine
$\theta$	I	$\sin\theta$	$\cos\theta$
$\pi - \theta$	II	$\sin\theta$	$-\cos\theta$
$\pi + \theta$	III	$-\sin\theta$	$-\cos\theta$
$2\pi -\theta$	IV	$-\sin\theta$	$\cos\theta$
$2\pi+\theta$	I	$\sin\theta$	$\cos\theta$

Example

Example 2. Obtain

\sin240\degree

\left(\frac{4\pi}{3} \text{rad}\right)

using the value of

\sin60\degree

\left(\frac{\pi}{3} \text{rad}\right)

Answer:

The angles

60\degree

and

240\degree

lie on separate quadrants yet they share the same reference angle of 60

\degree

. So let's apply the relationships mentioned in the table earlier.

\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt3}{2}

\sin\left(\frac{4\pi}{3}\right) = \sin\left(\pi + \frac{\pi}{3} \right)

\text{Since we know that:}

\sin\left(\pi + \theta\right) = -\sin\left(\theta \right)

\text{Hence},

\sin\left(\pi + \frac{\pi}{3} \right) = -\sin\left(\frac{\pi}{3}\right)

\text{Therefore, } \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt3}{2}

Applications of the unit circle

One of the primary uses of the unit circle is to evaluate and graph circular functions, such as sines and cosines. These functions are periodic, meaning that they repeat over a certain interval. The interval for sines and cosines is 360 degrees or 2π radians. The values of these functions can be determined using the coordinates of points on the unit circle. The interactive unit circle show right on top of this page demonstrates how the properties of a unit circle extend to periodic motions such as those of planets, pendulums, etc.

Question

Unit Circle Quiz. Complete the blank unit circle chart, using your knowledge of quadrants and the values filled on the chart.

Solution:

Here's how you'd work out the missing ratios for the angle

\frac{5\pi}{4}

rad.

Step 1. Identify the reference angle

\frac{5\pi}{4}

lies on the 3rd quadrant its reference angle is determined by subtracting it by

\pi

\frac{5\pi}{4} - \pi = \frac{5\pi}{4} -\frac{4\pi}{4}

\frac{5\pi}{4} - \pi = \frac{\pi}{4}

\text{Hence the reference angle is } \frac{\pi}{4} \text{ rad}

Step 2. Infer the desired coordinates from the reference angle.

\text{The coordinates of } \frac{\pi}{4} \text{ : } \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)

The magnitude of the x and y coordinates of

\frac{5\pi}{4}

would be identical but the signs would be negative as it lies in the 3rd quadrant.

\text{Therefore},

\text{The coordinates of } \frac{5\pi}{4} \text{ : } \left(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)

You can use a similar approach to figure out the rest of the missing coordinates and obtain the complete unit circle chart.