Why is it important to convert angle units when using trigonometric functions?

[fog37]

TL;DR Summary

Trig functions and their input argument

Hello,

Periodic trigonometric functions, like sine and cosine, generally take an angle as input to produce an output. Functions do that: given an input they produce an output.

Angles are numerically given by real numbers and can be expressed either in radians or degrees (just two different units). We know that $\pi/4$ and $45^\circ$ are the exact same angle even if they numerically different. How can I clearly explain that the function $sin(x)$ produces the same numerical result, i.e. $sin(\pi/4) = sin(45^\circ)$ even if, numerically, the inputs are the same?

A function, in general, produced a different numerical output if he input is provided a different unit. For example, the circumference $C=2\pi r$ will be different in $r$ is expressed either in $cm$ or $m$. But the trig functions don't...

Thanks!

 

[DaveC426913]

A function, in general, produced a different numerical output if he input is provided a different unit. For example, the circumference $C=2\pi r$ will be different in $r$ is expressed either in $cm$ or $m$. But the trig functions don't...

cm and m aren't really different units except inasmuch as the "centi" simply means 1/100th.

A 10cm circumference is the same length as a .1m circumference, just expressed in a more readable way.

 

[jedishrfu]

I think these can be demonstrated by the definition of 45 degrees as 1/8 of 360 degrees (degrees in a circle) just as $\pi/4$ is 1/8 of $2\pi$ radians (radians in a circle).

Hence the sin of this angle however it is represented must have the same value.

 

[fog37]

cm and m aren't really different units except inasmuch as the "centi" simply means 1/100th.

A 10cm circumference is the same length as a .1m circumference, just expressed in a more readable way.

True. I guess I should have used meters and yards.

 

[fog37]

I think these can be demonstrated by the definition of 45 degrees as 1/8 of 360 degrees (degrees in a circle) just as $\pi/4$ is 1/8 of $2\pi$ radians (radians in a circle).

Hence the sin if this angle however it is represented must have the same value.

Yes, the sine function is, in the context of a right triangle, a ration between one side and the hypotenuse.

What remains difficult to explain is that, conceptually the two angle measurements are the same but they are numerically different. The sine function seems to know that, under the hood, and produce the same output...

 

[jedishrfu]

Its also true that you can use a unit circle to geometrically compute the sin and cos of an angle. The fact that the angle is basically 1/8 of the circle is the same as saying its 45 degrees or $\pi/4$.

With respect to the sin() on your calculator always coming up with the same value that's just good programming on the calculator programmer's part. He/she likely converted all degree values to radians prior to computing the sin() via some mathematical series calculation which insures a good value for the sine.

 

[FactChecker]

There is no function $sin()$ where $sin(\pi/4) = sin(45^{\circ})$. There are two different functions, one which expects a radian input and another which expects a degree input. You will see this if you ever give the wrong input to a function. If we use subscripts to distinguish between the two functions, then you can say that $sin_{radian}(\pi/4) = sin_{degree}(45^{\circ})$

 

[jedishrfu]

ON the calculator you must choose what form of angular measure you will be using. Some famous and not-so-famous mistakes have been made someone put in the wrong input value.

I recall drawing a rosette plot for work in Java and using sin / cos to compute x,y values (polar coords to x,y coords) for the angular data. Instead of getting a nice squiggly circle, I got this mass of tangled lines.

It was then that I realized I forgot to convert my degree values into radian values since the sin()/cos() expected radian values. My 360 degree value was interpretted as 360 radians which is roughly circling the circle 60 times before plotting my x,y value.

Some of the famous ones:

https://www.bbc.com/news/magazine-27509559

 

[FactChecker]

ON the calculator you must choose what form of angular measure you will be using. Some famous and not-so-famous mistakes have been made someone put in the wrong input value.

I recall drawing a rosette plot for work in Java and using sin / cos to compute x,y values (polar coords to x,y coords) for the angular data. Instead of getting a nice squiggly circle, I got this mass of tangled lines.

It was then that I realized I forgot to convert my degree values into radian values since the sin()/cos() expected radian values. My 360 degree value was interpretted as 360 radians which is roughly circling the circle 60 times before plotting my x,y value.

Some of the famous ones:

https://www.bbc.com/news/magazine-27509559

I can second that. When you see wild scattered results from some geometry that you expected to be smooth, it is a good idea to look for degrees being input to a trig function that thinks it is radians.