Trigonometry is basically the study of angles and their relation to certain calculations. Therefore, it makes sense that the first thing we will review is the different classifications of angles and how they can be expressed.
What’s more, every single calculation you perform in trigonometry is going to be in relation to an angle of some sort. Thus, you should be able to visualize what these angles look like, especially as you progress through your studies.
Understanding angles on a fundamental level is important because at some point, you are going to need to find reference angles to solve certain problems, and you might need to know which quadrant an angle lies in. Of course, you can always refer to a unit circle to help with these problems, but if you have your key angles memorized, it makes it a lot easier to remember your unit circle.
For now, let’s focus on the basics. Angles are super easy and fun, as I’m sure you will see! Lets get into it.
6 Types of Angles
When referring to angles, they will usually be referenced to in terms of θ (theta), but sometimes you will α, X, or Y to reference an angle without a given a measure. For the purpose of clarity, I will be using θ.
**When angles are measured, they begin on one side and are measured counter-clockwise until they meet the terminal side (where the angle ends). If the angle is negative, the rotation is clockwise.
Obtuse Angles
90º<θ<180º
They are greater than 90º but smaller than 180º.
Acute Angles
0º<θ<90º
These angles are greater than 0º, but less than 90º.
(To find the functions of acute angles, click here.)
Right Angles
θ=90º
Right angles are always equal to 90º.
Straight Angles
θ=180º
Straight angles are always equal to 180º, and while they may not be what comes to mind when you think of angles, they are at least for trigonometry’s purposes.
Reflex Angles
180º<θ<360º
This is where it’s important to pay attention to the angle markers. The reflex angle could potentially look like an acute angle if you miss the rotation. That little circular line denotes the angle and rotation being shown.
Full Rotation
θ=360º
And as we can see, we have come full circle with the full rotation angles. Any angle that makes a circle, at 360º, is going to fall under this category.
Special Angle Relationships
There are three important angle relationships that will be useful as you apply trigonometry. Let’s dive in.
Supplementary Angles
Two angles are said to be supplementary if their sum is 180º.
Example: 90º+90º=180º. 128º+52º=180º.
If you are given the measure of one angle, and you need to find it’s supplement, you would simply subtract the given angle from 180º to find the supplementary angle.
Example: Find the supplement of 30º
180º-30º=150º
The angle that is supplementary to 30º is 180º.
Complementary Angles
Two angles are said to be complementary if their sum is 90º.
Example: 45º+45º=90º
Finding the complementary angle to any given angle is much the same as finding the supplement, except you will subtract the angle from 90º to find the complement.
Example: Find the complement of 15º
90º-15º=75º.
Co-Terminal Angles
Co-terminal angles end at the same point.
The angles differ by a multiple of 360º.
Example: 415º-360º = 55º
415º and 55º are co-terminal.
You can either add or subtract 360º as many times as you want to find different co-terminal angles for any given angle.
Degrees, Minutes, and Seconds (DMS)
While we can express angles in terms of their degrees (º), we can also express them in terms of degrees, minutes, and seconds.
1 degree = 60 minutes or 1º = 60′
1 minute = 60 seconds or 1′ = 60″
1 degree = 3600 seconds or 1º = 3600″
1^\circ = \frac{1}{360}\textnormal{ of complete rotation}
\\1'=\frac{1}{60}\textnormal{ of a degree}
\\1"=\frac{1}{60}\textnormal {of a minute or } \frac{1}{3600}\textnormal{ of a degree}Converting from DMS to Decimal Degrees
Say you are given the angle 53º15′, and you must convert it to decimal degrees. Since we know that 1′ is equal to 1/60th of a degree, we can use this ratio to help us convert. Set the minutes in a fraction over 60, and add it to the given 53º.
53^\circ15' = 53^\circ+(\frac{15}{60})^\circYou can simply use your calculator to perform the division, and plug in the given decimal after 53º.
The answer is 53.25º.
Converting from Decimal Degrees to DMS
In order to convert an angle from decimal degrees to DMS, you are going to perform the opposite function as converting from DMS to decimal degrees.
For instance, convert the angle 31.26º to DMS.
Since 1 minute is equal to 1/60th of a degree, we are going to multiply the decimal number by 60, and then plug that in for our minutes.
31.26^\circ=31^\circ+.26(60) \\=31^\circ+15'+.6(60) \\=31^\circ15'36"
.26 x 60 is equal to 15.6. Therefore, plug in the 15 for your minutes, and multiply the remaining decimal number by 60 again to get your seconds.
You end up with 31º15’36” as the decimal degrees 31.26º expressed in DMS.
Let’s Try a Word Problem
\textnormal{A tire rotates 900 times / min. Through how many degrees }
\\\textnormal{does a point on the edge of a tire move in }\frac{1}{3}\textnormal{ of a second?}To solve this problem, we are going to employ a simple ratio with the information we know.
1) The tire rotates 900 times in 1 minute.
2) 1 minute has 60 seconds.
Use this information to set up your first ratio.
\frac{\text{900 rotations}}{\text{1 minute}}\cdot\frac{\text{1 minute}}{\text{60 seconds}}The minutes will cancel each other out, so you can multiply across to get:
\frac{\text{15 rotations}}{\text{1 second}}\cdot\frac{1}{3}=\text{5 times}2) One full rotation has 360º , so we can multiply the resulting number by 360º for our final answer.
5\cdot360^\circ=1800^\circ\text{ in} \frac{1}{3}\text{of a second}Conclusion
Hopefully by now you are comfortable with the different classifications of angles, their relationships to one another, and converting between the different units of measurement. If you would like to watch a helpful video that explains these conversions, you can watch one here.