The application of trigonometric (trig) functions is widely used in our world. These functions are one of the basic math functions in areas like triangulation, which is used in criminal investigations and cell service. They're also used in navigation, surveying, computer graphics, and music theory. Did you know the shape of a vibrating guitar string is the same shape as the sine wave? This isn't a coincidence. In this lesson, we will be learning about the trigonometric function cotangent.
To learn about cotangent, we must first review what a tangent is. The tangent function can be found by comparing the opposite side over the adjacent side. This is also viewed by comparing the sine over the cosine functions. Using this triangle that you can see below, we can determine the lengths of the sides by comparing the sides based on the angle we're using.
Tangent is sine/cosine, or opposite/adjacent. We can see why by calculating using sine/cosine. You can see how these are calculated below.
This set up is important because cotangent is the reciprocal function of tangent. Cotangent is just the 'flipped' version of tangent, as you can see with the equation below.
Applications of Cotangent
Cotangent is used the same way the sine, cosine, and tangent functions are used. You can use them based on a right triangle, using the opposite and adjacent sides of the triangle, or you can use it based on the unit circle, which shows the angles in radians.
If we set up a 45-45-90 triangle, we can find the cotangent easily, as you can see below:
Notice the 45-degree angles have sides opposite of 1 and the hypotenuse is square root of 2. To find the cotangent of the 45-degree angle, it is adjacent/opposite, or 1/1, or 1.
Using the 30-60-90 triangle, the cotangent of the 30-degree angle is sqrt(3)/1, or square root of 3. The cotangent of the 60-degree angle is 1/sqrt(3).
If I wanted to find the height of a particular tree based on the shadow it throws when the sun is at a 30-degree angle, we can find this using cotangent. We'll use the 30-60-90 triangle to find this.
The cotangent of a 30-degree angle is square root of 3. Setting up our cotangent ratio, we can calculate the height of the tree this way:
If a contractor is standing 20 feet from a wall and is looking at a 45-degree angle to the top of the wall, he can find the height of the wall.
Unit Circle Used to Find Cotangent
The unit circle is another way to work with trigonometric functions. Instead of degrees, the unit circle is shown using radians, as you can see in the image below.
This unit circle shows both the degrees and radians. We can come up with these exact same answers using the unit circle.
The ordered pairs (x,y) can be viewed as (cos,sin). This is where the x-value represents the cosine and the y-value represents the sine of each degree or radian measure. Also remember, cotangent is the reciprocal of tangent, so if tangent can be represented as sine/cosine, then cotangent can be represented as cosine/sine.
Let's try two more using the unit circle. Find the cotangent of a 90-degree, pi/2 angle. Since the unit circle crosses at (0,1), the cotangent would be 0/1 = 0. At 180-degrees, the cotangent would be -1/0, which is undefined. This means that there is a vertical asymptote at the 180-degree point, or pi, on a cotangent graph. Below is a graph of the cotangent function:
Where there are vertical dotted lines, this indicates the points on the graph where the function cannot cross. This is where the function is undefined.
Let's take a few moments to review the important information that we learned. All trigonometric functions and identities are based on two functions, sine and cosine. The cotangent function is one of six basic functions in trigonometry. This is the reciprocal function of the tangent function. The tangent function is sine/cosine, so the cotangent function is cosine/sine.
When using the 45-45-90 triangle or the 30-60-90 triangle, the cotangent can be found by adjacent/opposite. When using the unit circle, the cotangent can be found by cosine/sine which is also x/y when working with the ordered pairs on a unit circle.
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