The Science That Draws Necessary Conclusions

I have made an assessment for what was taught so far. You can see it in the sidebar on the right or click on this. Unfortunately, I realized I forgot to mention the relevance of which quadrant the angle lies in. I will edit that into the special angles post. The answers will be posted in a few days. I am not yet sure where I will put them.

Before I create a test for the posts up to now, it is important for me to discuss coterminal and reference angles.

There are two parts of an angle, the initial side and terminal side. As the names imply, the initial side is the side where an angle begins and the terminal side is where the angle ends. The vertex is the point where the two sides meet. One can graph an angle on the coordinate plane. Standard position is when the vertex is at the origin (0, 0) and the initial side lies on the x-axis going in the positive direction. If the terminal side is moved counterclockwise, it is positive; if it is moved clockwise, it is a negative angle. Angles can be of any size, even greater than . An arrow normally notes the number of rotations, though it is not found in the picture above.

Coterminal angles are angles whose terminal sides lay upon one another but are not the same angle. For example, , , and are coterminal angles because their terminal sides lay on each other. The reference angle of an angle is the distance of its terminal side from the x-axis. Angles from to have a reference angle the same as the angle. So the reference angle of is .

The following only applies to angles between and . If the terminal side lies in the second quadrant, the reference angle is . If the terminal side lies in the third quadrant, the reference angle is . If the terminal side lies in the fourth quadrant, the reference angle is . The quadrantal angles, , , , and , have no reference angle.

To find the reference angle of other angles, it is easiest to find a coterminal angle between and . Simply add or subtract 360 until you find it.

For those who have graphing calculator, I made a program that you can use with it.

:Input ":", A
:
:While A>360
:A-360A
:End
:
:While A0
:A+360A
:End
:
isp "COTERMINAL:",A
:
:If (A=90 or A=180 or A= 270 or A=360)
:Then
isp "QUADRANT ANGLE"
:Stop
:End
:
:If (A>270 and A180 and A90 and A<180)
:180-AA
:
isp "REF ANGLE:",A


It may not be the best, but it works. You may want to add a “DelVar A” at the end. The next post should be announcing a Trig Test Part 1. The actual test will be on a page listed in the sidebar.

The special angles are , , and . could also be considered a special angle. The values of the trigonometric functions of these angles have specific values. It would be best to commit these values to memory. Consequently, this lesson is mostly memorization.

First, it is useful to know the radian measures of these angles. In radian measure, the angles are equal to divided by the number of times it goes into 180 since is equal to in radian. So is the same as , is the same as , is the same as , and is the same as . Multiples of these angles’ radian measures are found by multiplying the radian measure by then number of times the measure goes into it. For example, , which is equal to , has a radian measure of .

The rest is memorization. To my knowledge, there is no other option; however, only the sine and cosine functions need to be remembered. The other functions can be found based off of these as shown in the last lesson. Without further ado, the sine of is and the cosine is . Because sine and cosine are reciprocal functions, the values for are reversed. So the sine of is and the cosine is . Also, the sine and cosine functions for are the same. Both equal which is rationalized.

Remember that the cosine of the angle is the x-coordinate and the sine is the y-coordinate on the unit circle.

Added on April 6, 2008:

Angles whose reference angles is a special angle will also have these trigonometric values; however, they may be positive or negative depending on what quadrant it lies in. Since sine is related to the y-coordinate, it will be positive when the angle lies above the x-axis and negative when below. Since cosine is related to the x-coordinate, it will be positive when the angle lies to the right of the y-axis and negative when it lies to the left. As always, the other functions depend on these two.

Besides sine and cosine, there are four other trigonometric functions. They are tangent, secant, cosecant, and cotangent. Each can be found using sine and cosine.

Tangent is equal to the sine of an angle divided by its cosine, or . The cosecant, secant, and cotangent functions are the reciprocals of sine, cosine, and tangent respectively. This means , , . These functions are also ratios between the sides of a right triangle.

If you remember (TOA from SOH-KAH-TOA), tangent is the length of the opposite over the adjacent. So, . Because the other functions are the reciprocals, the fraction is simply flipped over. For example, the cosecant, the reciprocal of sine, is equal to the ratio of the hypotenuse over the opposite side. Therefore, , , .

Unfortunately, there is no phrase to help you remember which is which. It is easiest to remember what the functions are reciprocals of. Also, note that the functions are not continuous. Any function where the denominator is zero is undefined. The next post will be about special angles.

There are six trigonometric functions. The two most important ones are sine and cosine. The others can be derived from these two functions. The trigonometric functions are based on the ratios between sides that exist in right triangles.

Sine is the ratio of the side opposite of to the hypotenuse. Therefore, . Cosine is the ratio of the side adjacent to to the hypotenuse. Therefore, . A common mnemonic device to remember which funtion refers to which rations is SOH-CAH-TOA. SOH means ; COS means . TOA is for the function tangent. For now just know TOA stands for . Sine and cosine are cofunctions. That means that as well as .

The unit circle is another way to look at sine and cosine. Any angle passes through one point on the unit circle. The point the angle passes through is . Remember that sine and cosine functions both equal the length of a side over the hypotenuse. The length of the hypotenuse of the triangle formed from the angle will always be one in the unit circle. The side opposite of the angle is the same the y-coordinate of the point and the side adjacent is the same as the x-coordinate.

Sine and cosine are also related to the x and y coordinates where an angle intersects a circle concentric with the unit circle. The difference is that the hypotenuse is no longer equal to one. Instead, it is the radius of the circle. Therefore, and .

Next to be discussed are the remaining four trigonometric functions. They should be fairly simple to learn if sine and cosine is easy. The animation above shows the graph of . It will be a topic of later posts.

Before I write about trigonometry, I would like to explain the measurement of angles. More people would think of a circle as having than . The measurements are in degrees and radians respectively. Most people should already have a good understanding of degrees. Fortunately, understanding radian measurement is also easy.

A circle is made up of 360 degrees. A after a number means the the number is in degrees . While decimals are most often used, minutes and seconds can be used to represent an angle. A minute is of a degree and a second is of a minute ( of a degree). Minutes and seconds are denoted by ‘ and ” respectively. For example, 27′ 36″.

More advanced math almost always uses radians. It is likely that you will eventually prefer to use radian measurement.

Travel along the circumference of a circle the distance of its radius. The angle formed is equal to one radian. It is equal to about . Radians are often measured in terms of . In this form, one radian is . Conversely, it can be written as or .

To convert degrees to radians, multiply the number by . To convert radians to degrees, multiply the number by . That is it for now. Sine and cosine will probably be the topic of the next post.