HSC Mathematics Advanced, Extension 1, and Extension 2/Trigonometric functions
Introduction to Radians
[edit | edit source]Throughout high school mathematics, you have mostly likely being working with degrees. Degrees are used to measure the size of an angle. Likewise, radians are also used to measure the size of an angle.
Definition Radians:
π radians = 180°
What is the purpose of radians in contrast to degrees if they can both be used to represent angles?
As it turns out, the formula for gradient of the trigonometric function became more complicated when radians were used. By defining radians such that π radians = 180°, the formulas for the gradient became simpler.
We will state the following theorems without proof:
(We will make reference to the trig functions, sin and cos. If you are unfamiliar with trig functions, that is okay as we will go over the definitions of the trig functions later in the course)
Theorem 1: Gradient of sin(x) where x is in degrees
gradient of sin(x) = π/180*cos(x)
Theorem 2: Gradient of sin(x) where x is in radians
gradient of sin(x) = cos(x)
Notice how the π/180 disappears in theorem 2 when x is in radians. This simpler formula made radians much more preferable to use over degrees.
How do we convert from degrees to radians and radians to degrees?
Now we know why radians are important, you may be interested in how to convert degrees to radians.
Algorithm 1: Conversion of Degrees to Radians
Let θ° to be value we want to find in radians.
- Use the definition. From the definition, we know 180 degrees = π radians.
- Divide both sides by 180, this lead us with 1 degree = π/180 radians
- We multiply both sides by θ, this leads us with θ° = θ*π/180 radians.
It should be clear why the algorithm describe above works, as only the definition of radians and basic arithmetic is being used.
Example 1: Find 88.96° in radians.
180° = pi radians
1° = π/180 radians
88.96° = 88.96*pi/180 radians
Shortcut: You may notice in step 3 of algorithm 1 we have θ° = θ*π/180 radians. Instead, step 1 and 2 of algorithm 1 we can just apply step 3.
If we replace θ with 88.96 in step 3 we end up with 88.96° = 88.96*π/180 radians, which is the same equation as below. In general, if you find any unit in degrees to radians, all you have to do is multiply by pi/180.
Example 2: Find 50° in radians
50 degrees = 50*π/180 radians
Algorithm 2: Conversion of Radians to Degrees
Let θ radians to be value we want to find in degrees.
- Use the definition. From the definition, we know π radians = 180 degrees.
- Divide both sides by pi radians, this lead us with 1 radian = 180/π degrees
- We multiply both sides by θ, this leads us with θ radians = θ*180/π degrees
Example 3: Find 3.2 radians in degrees
π radians = 180°
1 radian = 180/π°
3.2 radians = 3.2*180/π°
Shortcut: You may notice in step 3 of algorithm 2 we have θ radians = θ*180/π°. So whenever we want to convert in radians to degrees, all we have to do is multiply by 180/pi
Multiplying 3.2 radians by 180/π we end up with 3.2*180/π°. In order words 3.2 radians = 3.2*180/π° which is the same result we got when applying algorithm 2.
Important Note: In this section so far, we have being indicating a unit is in radians by writing radians next to the number. In practice, this will not be done. E.g. π radians will simply be written as π.
How can we tell if a number is in degrees or radians?
If you see a superscript circle i.e., ° this will indicate the unit is in degrees. Although there are symbols that can be used to indicate a unit in radians, in practice no symbol is used. Generally speaking, through the context, you should be able to determine if a unit is in degrees or radians. Any calculations involving the gradient will mean that radians are likely being used, as you can remember in our discussion from the section: "What is the purpose of radians in contrast to degrees".