Why use sin for x and cos for y when resolving a vector in quadrant 3

[th77]

I have a problem that includes an acceleration vector 'a' located in the 3 quadrant and it makes an angle (theta) 30 degrees to the Negative Y axis.
The solution manual shows the the vector resolved like
a = (12.0 sin 30 m/s^2) i - (12.0 cos 30 m/s^2) j
Why take the sin of 30 for the x direction and the cos of 30 for y? I've always done it cos of x and sin of y.

 

[vanesch]

I have a problem that includes an acceleration vector 'a' located in the 3 quadrant and it makes an angle (theta) 30 degrees to the Negative Y axis.
The solution manual shows the the vector resolved like
a = (12.0 sin 30 m/s^2) i - (12.0 cos 30 m/s^2) j
Why take the sin of 30 for the x direction and the cos of 30 for y? I've always done it cos of x and sin of y.

Because the angle is here defined differently than usual...

 

[whozum]

The reason you flip them is because the reference angle is made with the y-axis instead of the x. Usually we measure the angle from the x axis.. The reason it works when you flip then is due to the two properties of sin and cos:

$\sin(x-90) = \cos(x)$

$\cos(x-90) = \sin(x)$

 

[th77]

The reason you flip them is because the reference angle is made with the y-axis instead of the x. Usually we measure the angle from the x axis.. The reason it works when you flip then is due to the two properties of sin and cos:
$\sin(x-90) = \cos(x)$
$\cos(x-90) = \sin(x)$

Thanks! That leands me to another question...
In this problem, the angle is 30 degrees with the negative axis so shouldn't cos 240 be equal to sin 30? They come to -0.5 and 0.5 respectively.

 

[vanesch]

Thanks! That leands me to another question...
In this problem, the angle is 30 degrees with the negative axis so shouldn't cos 240 be equal to sin 30? They come to -0.5 and 0.5 respectively.

You're turning in the wrong direction: 30 degrees starting from the negative Y axis (=270 degrees) gives you 300 degrees, not 240...

 

[th77]

You're turning in the wrong direction: 30 degrees starting from the negative Y axis (=270 degrees) gives you 300 degrees, not 240...

the vector is in the 3rd quadrant so isn't it 240?

 

[whozum]

$\cos(240)$ measures the (60 deg) reference angle from the negative x-axis in the third quadrant, where cosine is always negative. However if you are going to refer to the angle from hte positive x axis, then you are also taking $\sin(240)$ and not $\sin(30)$. Those relations are technically supposed to be used in the first quadrant, or just as a relative measure.