I’m a senior in high school and I don’t know how to choose sin or cos in physics problem. For the most part I memorised them. For example, why is the flux of a magnetic field = BAcos(angle) and why is the electromagnetic force = ILBsin(angle).
The last two years weren’t normal as we studied online so it’s highly appreciated if someone can eli5 and fill in the gaps for me.
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In these cases you are dealing with sin and cos because you are essentially comparing the magnitude of two vectors.
As a simpler example, consider a cube sliding down a wedge. Gravity creates a directional force straight down. But the object can’t go straight down because of the wedge. So wedge essentially redirects that force in a different direction by some angle.
From these two vectors/directions we can construct a right-angled triangle with the primary force (gravity) as the hypotenuse and one of the legs being the wedge itself. What we want to know is the length (magnitude) of the leg of the triangle that corresponds to the wedge.
Illustration:
[https://i.imgur.com/Ljn09Wa.png](https://i.imgur.com/Ljn09Wa.png)
For this we would use sin. Why? Because:
sin(theta) = opposite/hypotenuse
And the side opposite the angle, theta is what we want to figure out. Since we know the angle of the wedge (theta) and the force due to gravity (Fg) we can figure out the opposite side (net force) using sin.
If, instead, we measured the angle of the wedge from the other angle, we’d use cos, because then the “net force” side would be adjacent to the angle.
So the choice of cos vs. sin depends on how the different vectors (directions of force) relate to teach other.
Sine and Cosine are functions that describe the ratios of the lengths of different sides in a right-angle triangle. If you draw a right-angled triangle, pick one of the other angles, Sine tell you about the ratio of the side opposite that angle to the longest side, Cosine tells you about the ratio of the side next to the angle to the longest side.
So they are really useful when doing anything involving triangles.
What we’re doing in electromagnetism is slightly different, though. The 19th century physicists working on electromagnetism had to invent a whole new area of maths to help understand this area, and unfortunately we tend not to teach it to you in school. So you are trying to solve problems that the people who developed this area needed new maths to solve, but without that maths. Which is why it is a bit confusing and a bit of a fudge.
Anyway… Generally, what is going on when we have sines and cosines is that we have two “things” (vectors) that are pointing in different directions, with some angle between them. Our basic formulae assume that either those things are in the same direction, or are at right-angles (flux = BA, or EM Force = ILB). If they aren’t, we need to add in a scale factor to account for the angle. That is either going to be cosine or sine as these essentially tell us how much of one vector is in the two different directions compared with the second vector (either in the same direction, or in the perpendicular direction); this is why it comes up a lot when resolving forces in mechanics. Which one we need will depend on which angle we look at, and what we want to do.
For example, if you have the angle between two vectors, and you want to know how much of one is in the same direction as the other, you will need cosine. If you have the angle between two vectors and you want to know how much of one is perpendicular to the other you will need sine [and that’s what we’re using in F = ILB sin(angle)].
Similarly, if you have the other angle in the right-angle triangle (where one side is one of your vectors, and the longest side is the other), you’d need sine if you wanted how much of them is in the same direction, and cosine if you wanted how much is in the opposite direction [which is our flux = BA cos (angle)].
Vectors make this all a lot easier. Generally the sine comes up when you have a vector “cross” product, and cosine when you have the vector “dot” product (these are the two ways of multiplying vectors together). Although sometimes differentiation is involved, which changes sines into cosines, and vice versa. If you are not comfortable with vectors, or trig geometry in general, you’ll probably need to just memorise. Explaining them in detail requires a lot of diagrams, and Reddit isn’t the best place for that.
Go back to the original diagrams of right-angled triangles where you first learned about trigonometry functions. Sin = opposite side over hypotenuse, cos equals adjacent side over hypotenuse.
Formulas are usually in the form X = K.trig(angle) where K is something you know, B*A in the first example. You need to find out X which is in a direction at some angle to K so draw a diagram where K is the hypotenuse and work out whether the X you want is opposite or adjacent the the angle you know.
Another way is to remember that sine is zero at 0 and 180 degrees (π radians) and maximum at 90 and 270. Cosine is opposite. So does your X increase or decrease as the angle changes from the direction of X being aligned with K to being at right angles to it?
The simple answer – what is the value with an angle of 0? If the value is low when the angle is zero, the formula will use cosine. If the value is high when the angle is zero, you’ll use sine
The magnetic flux is a number (scalar) that tells you how much magnetic field lines are going through a surface.
Say for example that there is a uniform magnetic field present in the z-direction in your room (so the magnetic field is pointing upwards from the ground in standard 3D space). Say you have a sheet of paper that is lying on the ground. You then have a magnetic flux going through the sheet of paper that is equal to the product of B and A, where B is the magnetic field strength and A is the area of the paper. You can imagine this as the amount of magnetic field that your sheet of paper is “catching”.
If you now hold the paper on its side, none of the magnetic field lines can go through the sheet. The sheet cannot “catch” any magnetic field lines since they are moving parallel to the sheet. The flux is thus equal to zero.
Since the direction of both the magnetic field lines and the sheet of paper are important to determine the flux, the orientation of the field lines with respect to the paper needs to be known in order to calculate the flux. Because it is easy to do so, we have decided to give the orientation of an area by its normal vector. The normal vector is simply the vector that is perpendicular to the sheet.
Like I said, the flux gives a number and is the result of two vectors, namely the magnetic field strength B and the area A. The flux is then simply defined to be the inner product of B and A, which becomes:
Psi = B*A*cos(alpha), where Psi is the flux and alpha is the angle between the magnetic field lines and the normal vector.
So you may be familiar with the idea that E and B are vectors, not scalars. Whenever you multiply two vectors together, you have a choice of two ways to do it: the *scalar* product (what component of vector 1 is in the direction of vector 2), and the *vector* product (uh… I can’t explain this to a 5yo, it’s basically ‘find a vector at right angles to these two whose length is determined by their lengths’).
The scalar product has a cosine in it and produces a scalar. The vector product has a sine in it and produces a vector.
So the flux, a scalar, is a scalar product with a cosine in it. The force, a vector, is a vector product with a sine in it.
The easy way to remember is to imagine what you’d expect if the angle were 0 and what you’d expect if it were 90.
Sin(0) = 0, Sin (90) = 1
Cos(0) = 1, Cos(90) = 0
The amount of flux is based on the field going _through_ the surface. Remember that the vector associated with the surface is perpendicular to the actual surface. So when the angle between A and B is zero, the surface is completely perpendicular to the field, so that is when the flux is at its maximum. When the angle between A and B is 90 degrees, that means the surface is parallel to the field, and none of the field goes through the surface.
So flux is a maximum at 0 degrees, and minimum at 90. So you use Cosine.
For electromagnetic force, it’s the opposite. The most force happens when the angle is 90, and there’s no force when the charge is moving parallel to the field. So maximum at 90, minimum at 0 – use Sine.
here’s what you need to know without getting too deep into math.
sin and cos are used to select out the up/down, or the left/right pieces of a particular vector. could be force, velocity, whatever. anything that is a vector. this is done because maybe you don’t care about how fast the object is moving up/down and you only care about the left/right motion.
you can quickly see which one you need with intuition. you can look at an object going down a small incline (classic first-year physics). most of the force of gravity will apply directly down to the surface (which will balance to 0 Newtons with the normal force), and a little bit of force will be directed down the incline to move the object.
you can try both sin and cos in your calculator to check which one gives you a small force or a big force. the small one will be the one going downhill, and the big one will be the one pushing directly into the plane to be balanced out with the normal force.
you should know by memory that sin of a small angle is close to 0 and cos of a small angle is close to 1.
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