jpd5184 said:
so if the vector is in the first quadrant and i am using the arctan. which is cos/sin
Usually (but not necessarily always), the angle
θ is defined with respect to the positive x-axis.
If it is, the y-component is associated with the sine and the x-component the cosine. But The angle
θ is
not always defined with respect to the positive x-axis in
all problems all the time. So sometimes you'll need to figure this stuff out on a problem-by-problem basis.
But here is one thing you can and should memorize. The following is always true for all right triangles: In a right triangle where
θ is one of the angles (one of the acute angles -- not the 90
o one),
\sin \theta = \frac{\mathrm{opposite}}{\mathrm{hypotensuse}}
\cos \theta = \frac{\mathrm{adjacent}}{\mathrm{hypotensuse}}
\tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}}
and finally the Pythagorean theorem,
\left( \mathrm{hypotenuse} \right)^2 = \left( \mathrm{opposite} \right)^2 + \left( \mathrm{adjacent} \right)^2
Those are the really important ones to memorize.
Now draw a vector with its tail at the origin and its head in the first quadrant. Draw
θ as being the angle between the vector and the x-axis. Actually draw this on paper. Draw a dotted line from the tip of the vector, straight down to the x-axis (the dotted line should be parallel to the y-axis, and the dotted line should intersect the x-axis). You've made a right triangle. Now examine your drawing carefully and determine which component is opposite and which component is adjacent.
and cos and sin are both positive in the first quadrant then the answer would be positive.
so the arctan(50/60)= 39.8(degrees)
Yes, that's right!
And if you know the vector is in the first quadrant, you can trust that your calculator will give you the correct answer. But you can't trust your calculator to give you the correct, final answer if the vector is in any other quadrant.
I caution you here that things can get a little tricky with arc tangents, arc sines, etc., when dealing with multiple quadrants. If you're just given the task to calculate the arctan of a negative number on your calculator, the result will be a negative number, but you can't tell if it is in the second or fourth quadrant by that information alone. If you are asked to take the arctan of a positive number, you will get a positive number, but you don't know if the vector is in the first or third quadrant (without more information).
So as part of the process you need to keep in mind which quadrant the vector is into begin with. Then, after you use your calculator to get the arctan, adjust the answer accordingly to put it in the correct quadrant (which might involve negating the angle, and may also involve adding or subtracting 180
o.)
But you don't need to take my word for what to do in each particular quadrant. I encourage you to start drawing vectors on paper (with the tails at the origin), and figuring this out for yourself. About all you really
need to memorize are the formulas listed above. And don't forget that your calculator will assume you are working with simple right triangles with acute angles. You can figure out the rest (i.e. the rest of the geometry that your calculator does
not consider).
[Edit: although you can figure it out for yourself, you
will need to practice it. So please, take out some graph paper for this.
]