How do you determine the angle?

Question

Determine the exact angle(s) of theta if cos theta = -3/root2, where 0</=theta</=2pi.

Answer 1

Are you sure the question is not stating { cos theta =  -(3^.5)/2 }?

 

If that is so, then the question has an answer. If so, then the simplest way to proceed is to draw a circle out and recall when cosine is negative on the unit circle and at what angles does cos theta = -(3^.5)/2.

 

• In this case, cosine is negative in the 2nd and 3rd quadrant, meaning theta is between pi/2 and 3pi/2.
• If you recall from a 30 - 60 - 90 triangle, cos pi/6 gives us (3^.5)/2.
• This means that to get -(3^.5)/2, we have to have angles of pi/6 in the 2nd and 3rd quadrant.
• This gives us the two angles of 5pi/6 and 7pi/6, which are 150 and 210 degrees respectively.

Answer 2

If you attempt to use inverse cosine on both sides of cos theta = -3/root2, you get:

 

cos theta = -3/root2

cos^-1 (cos theta) = cos^-1 (-3/root2)

theta = cos^-1 (-3/root2)

 

If you plug cos^-1 (-3/root2) into your calculator, you will get an error.

 

This is because cos is normally between the numbers -1 and 1.  If the number is less than -1 (as in this case), you will not be able to find an angle.

 

If the problem asks specifically for cos theta = -3/root2, then you have no exact angle.  However, if you arrived at cos theta = -3/root2, then I advice reviewing your work.

 

If you have any questions, feel free to ask!