Find cos theta if tan theta= -square root 3/3 and falls in quadrant 2

Question

A. cos theta=-1/2

B. cos theta=1/2

C. cos theta=square root3/2

D. cos theta=-squareroot 3/2

E. none of these 

Answer 1

Hi Tam!

 

A picture is always better than words for these types of problems.  Draw an x-y axis and then draw an arrow, starting at (0,0) into the 2nd quadrant (up and left).    Theta is the angle made with one side being the negative x-axis and the other side of the angle being your newly drawn arrow.

 

In the second quadrant all y-values are positive and all x-values are negative.  So the tan(theta) = y/x = √3/(-3) would be more accurate.    (The fraction is negative, but it is because the y (top) is positive and the x (bottom) is negative.)

 

From the tip of your arrow in the 2nd quadrant, draw a vertical line down to the x-axis  (this should land on a negative x-value at the axis).  The length of this vertical line is the sqrt(3) from your tangent  fraction.      

 

Now, mark the horizontal section along the negative x-axis that ends with your vertical line as the "-3" from the tangent fraction.    

 

 

What you should now have is a right triangle.  The right angle should be on the negative x-axis, the hypotenuse should be an up-and-left pointing arrow from the origin.

 

Now you can figure out the length of the hypotenuse by a² + b² = c²

 

(√3)² + (3)² = 3 + 9 = 12 = c²,   so c = √12 = 2√3

 

Now you have a right triangle with theta between the negative x-axis and your arrow.  Opposite of theta is the sqrt(3).

Adjacent to theta is the (-3), and the hypotenuse is 2√3.

 

Now you can easily find the cos(theta).  Cosine is adj/hyp.  Adj = (-3) and Hyp = 2√3

 

cos(theta) = -3/2√3

 

Since math people like to rationalize denominators, multiply top and bottom by sqrt(3)

 

Now cos(theta) = (-3)(sqrt(3)/6

 

This will simplify to   -sqrt(3)/2

 

Looks like the final answer is D

 

 

Hope this helped.  :-)

Vicki