Why Do Cosine and Sine Give Different Angles for the Same Vector?

[k_squared]

Homework Statement

Find the magnitude of a and the smallest positive angle theta from the positive x-axis to the vector OP that corrosponds to a.

a= (3,-3)

Homework Equations

a₁ = ||a|| * cos theta
a₂ = ||a|| * sin theta
3. The Attempt at a Solution

||a|| = $$\sqrt{}$$(9+9) = 3$$\sqrt{}$$2
a₁ = 3$$\sqrt{}$$2 * cos theta

=1/$$\sqrt{}$$2 =acos = 45 degrees.

However doing the other side of the equation...

a₂= -1/$$\sqrt{}$$2 = asin = -45 degrees = 315 degrees, which is the right answer. I thought they were supposed to be consistent...

 

[Mark44]

Homework Statement

Find the magnitude of a and the smallest positive angle theta from the positive x-axis to the vector OP that corrosponds to a.

a= (3,-3)

Homework Equations

a₁ = ||a|| * cos theta
a₂ = ||a|| * sin theta
3. The Attempt at a Solution

||a|| = $$\sqrt{}$$(9+9) = 3$$\sqrt{}$$2
a₁ = 3$$\sqrt{}$$2 * cos theta

=1/$$\sqrt{}$$2 =acos = 45 degrees.

However doing the other side of the equation...

a₂= -1/$$\sqrt{}$$2 = asin = -45 degrees = 315 degrees, which is the right answer. I thought they were supposed to be consistent...

arccos produces an angle between 0 and 180 degrees, while arcsin produces an angle between -90 and + 90 degrees. Since your vector is in the fourth quadrant, an angle of 45 degrees wouldn't be right, nor would -45 degrees, since the problem asks for a positive angle.