NEED HELP WITH A QUESTION ASAP

[AsianNerd]

Hey im really confused and havnt seen a question like this before... it is the vcaa 2012 methods (cas) paper exam 2,, multiple choice question 19..
http://www.vcaa.vic.edu.au/Documents/exams/mathematics/2012/2012mmcas2-w.pdf
What does it mean by even? as stated in the solutions and itute??
and how was this answer reached?/
Thankyou please... most detailed answer possible!!

[Limista]

hi,

If we take A: sin(a), where a = theta, and apply sin(a) to the equation: 

f (pi – θ ) = –f (θ ) and f (pi – θ ) = –f (–θ )

sin(pi - a) = -sin(a)

If you think about the Cartesian plane, this statement is wrong. This is because sin(pi-a) is in the second quadrant, where sine is positive. So, the LHS is not the same as -sin(a), which implies that sine is negative. We can get rid of option A.

If we take B: cos(a), where a = theta, and apply it to the equation:


f (pi – θ ) = –f (θ ) and f (pi – θ ) = –f (–θ )

Take into consideration the Cartesian plane again.
cos(pi-a) is the same as -cos(a), because cosine is negative in the second quadrant.

cos (pi - a) is also the same as -cos(-a) = -cos(a).

Remember this rule -->   cos(-a) = cos(a) 

So the right answer should be B.

[AsianNerd]

Thanks very much... but what does it mean by even as stated in the answers?? also is is -cos(-a) in the second or third quadrant and why?

[Conic]

Thanks very much... but what does it mean by even as stated in the answers?? also is is -cos(-a) in the second or third quadrant and why?

http://en.wikipedia.org/wiki/Even_and_odd_functions

[Limista]

also is is -cos(-a) in the second or third quadrant and why?

well -cos(-a) = -cos(a).

-cos(a) can be located in the quadrants where cosine is negative. So, this is the 2nd and 3rd quadrants.

[AsianNerd]

please i know i sound stupid but how does -cos(-a) = -cos(a)?

[Sanguinne]

cos(-a) is located in the 4th quadrant. In this quadrant cos is positive therefore cos(-a) = cos(a)

[BubbleWrapMan]

please i know i sound stupid but how does -cos(-a) = -cos(a)?

Think of the graph of . What happens if you reflect it in the -axis? It stays the same. So .

[AsianNerd]

I get it now... Thanks guys!