241° is between 180° and 270° which is the third quadrant, QIII.  That's
the lower left quadrant.  241° has this terminal side and the red arc
shows its counter-clockwise rotation from the right had side of the x-axis.

  

Notice that 241° is made up of two angles of rotation. The 180° rotation 
(in green) and the rotation after 180° which is 241°-180° = 61° (in blue):



So the angle indicated by the blue arc, between the x-axis and the terminal
side is 61°, and we say that "the reference angle of 241° is 61°". This
REFERENCE angle has the same sine, cosine, and tangent as 241² except for
their positive and negative signs. [There is a problem here because "sine" and
"sign" are pronounced the same. So we'll say "positive-ness and negative-ness"
to avoid confusion between the two words when we talk about the positive-ness or
negative-ness of the trig rations and functions.]  

Now learn the sentence:  "All Students Take Calculus" to remember "All S T C"
For the quadrants, think of this as though it were an x,y coordinate system:

                    S|A
                    T|C

The major three trig functions are Sine, Cosine, and Tangent, the others
are their reciprocals.  This memory device only works with the major three
trig functions, Sine, Cosine and Tangent.

"All" reminds us that ALL trig functions are POSITIVE in the first quadrant QI.

"STUDENTS" reminds us that of the major three trig functions, the SINE is
POSITIVE in the second quadrant QII, and the other two major trig functions are
NEGATIVE there.

"TAKE" reminds us that of the major three trig functions, the TANGENT is
POSITIVE in the third quadrant QIII, and the other two major trig functions are
NEGATIVE there.

"CALCULUS" reminds us that of the major three trig functions, the COSINE is
POSITIVE in the fourth quadrant QIV, and the other two major trig functions are
NEGATIVE there.

If the major trig function, the SINE, is positive or negative in a quadrant, its
reciprocal, the COSECANT is also positive or negative there.

If the major trig function, the COSINE, is positive or negative in a quadrant,
its reciprocal, the SECANT is also positive or negative there.

If the major trig function, the TANGENT, is positive or negative in a quadrant,
its reciprocal, the COTANGENT is also positive or negative there.

Now if we have learned that, let's go through what this information tells us
about the positive-ness or negative-ness of cos(241°).

The cosine is negative in the third quadrant, QIII. But the cosine of the
reference angle is always positive, because if place in proper position it
would always be in the first quadrant QI where all trig rations and functions
are positive.  So to get the answer to this problem, simply put a negative
in front of cos(61°) and that's your answer:

Answer: cos(241°) = -cos(61°)

Edwin