How do you know if you need cos or sin?

[Abdul.119]

When solving a problem about shooting a cannon ball, why is the velocity in the x direction multiplied by cos and velocity in the y (vertical) direction multiplied by sin? like in the last example here http://www.physicstutorials.org/hom...rboat-problems/23-projectile-motion?showall=1

 

[phinds]

When solving a problem about shooting a cannon ball, why is the velocity in the x direction multiplied by cos and velocity in the y (vertical) direction multiplied by sin? like in the last example here http://www.physicstutorials.org/hom...rboat-problems/23-projectile-motion?showall=1

Do you understand vectors? Can you break a trajectory down into its coordinate component vectors?

 

[Abdul.119]

Do you understand vectors? Can you break a trajectory down into its coordinate component vectors?

Yes, I understand that in this type of problem the projectile is moving in two dimensions, so we use x for motion in the horizontal and y for the motion in the vertical direction, but not sure why horizontal is always associated with cos and vertical is associated with sin in this problem

 

[A.T.]

http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter02/section02.html

 

[DrClaude]

Yes, I understand that in this type of problem the projectile is moving in two dimensions, so we use x for motion in the horizontal and y for the motion in the vertical direction, but not sure why horizontal is always associated with cos and vertical is associated with sin in this problem

Because the angle is expressed with respect to the horizontal.

 

[sophiecentaur]

This sin / cos quandry is very common. I have a quick and dirty way to resolve the question which can often make more intuitive sense than strictly keeping to the Maths.
Sin(x) increases as x increases from 0 to 90 and cos(x) decreases. If you look at the mechanical situation (whatever it happens to be), it is vey often possible to decide whether the effect increases or decreases with the angle and that will (can) give you an inkling about which function to use.
There's a caveat here. Tan(x) also increases as x increases from zero and it is possible to take the 'wrong' two sides of your triangle. But as long as you are involving the hypotenuse, my method will help you to feel a bit more confident about your choice.

 

[Merlin3189]

I like Sopie's idea, but I also think visually:
If you look at a right angle triangle with a horizontal base and vertical side, the slope of the hypoteneuse is the angle between it and the horizontal, the vertical side is proportional to the sine of that angle and the horizontal side is proportional to the cosine.
The hypoteneuse represents the vector (motion or force for eg,) and the horizontal and vertical sides (always less than or equal to the hyp) represent the x and y components of the vector.
Without your calculator you can even use scale drawing to measure, roughly, the components of a vector, by drawing such a triangle.

 

[Dale]

My approach is somewhat similar to @sophiecentaur. I think, "what would happen if the angle were 0". In that case it should give either a 1 or a 0. If it gives 1 then it is cos and if it gives 0 then it is sin.

 

[robphy]

Before getting into the limiting cases, I prefer to first remind the student of the definitions by stating...
[in a right triangle] "cos goes with adjacent" [and "sin goes with opposite"],
sometimes followed by a comment ( like @DrClaude 's ) that the x-axis is often but not always chosen as horizontal.

 

[pixel]

Just start with the definitions: sin is opposite over hypotenuse, cos is adjacent over hypotenuse. So in the diagram below,

sinθ = Fy / F → Fy = F sinθ
cosθ = Fx / F → Fx = F cosθ

 

[pixel]

Note that for this diagram,

sinφ = Fx / F → Fx = F sinφ
cosφ = Fy / F → Fy = F cosφ

So the x-component is not always associated with the cos of the given angle, nor the y-component with the sin. After a while, however, it kind of becomes automatic if you just always refer to the definition of sin and cos.

 

[2milehi]

Before getting into the limiting cases, I prefer to first remind the student of the definitions by stating...
[in a right triangle] "cos goes with adjacent" [and "sin goes with opposite"],
sometimes followed by a comment ( like @DrClaude 's ) that the x-axis is often but not always chosen as horizontal.

SOH - Sin θ = Opposite/Hypotenuse
CAH - Cos θ = Adjacent/Hypotenuse
TOA - Tan θ = Opposite/Adjacent

Learned that in 10 grade, 32 years later it still sticks with me.

 

[sophiecentaur]

SOH - Sin θ = Opposite/Hypotenuse
CAH - Cos θ = Adjacent/Hypotenuse
TOA - Tan θ = Opposite/Adjacent

Learned that in 10 grade, 32 years later it still sticks with me.

In not-so-PC days, the legendary Mr Worthington told us of a Red Indian Chief, called SOH-CAH-TOA.
We all know the formulae,pretty well when given a proper looking right angled triangle, the right way up.
But the problem we all have (some more than others) is when the triangle is elusive and ti's not clear which is the hypotenuse and which is the 'next longest side' brings on the pains. It can be a great help to get as far from the Maths as possible and look at the thing 'mechanically', in fact, in the way that PF (myself included) tends to discourage in many cases.

 

[CWatters]

They still teach that in the UK but no native people are harmed in the making...

 

[houlahound]

The greater confusion is there is nothing sacred about either sin or cos if you consider they give the same answer when they are correctly phase shifted.

 

[sophiecentaur]

The greater confusion is there is nothing sacred about either sin or cos if you consider they give the same answer when they are correctly phase shifted.

Yes but anyone who can handle that will not be having trouble with a triangle of forces, will they?

 

[houlahound]

Good point, I just feel its pedagogically misleading, geometry aside, to make like they are different things when they are just different starting points on the same thing.