Thank you, darkz!
I really appreciate it.
I was wondering how you recognise complementary relationships for circular functions?
Are they present when they are in the form pi/2 and 3pi/2 minus/add a variable?
Rote learning doesn't work for these as you're far too prone to make a mistake. Rather, you need to know what cosine actually means.
Cosine = complementary sine, which means sine of the complementary angle, and the complement of any angle \(\theta\) is \(\frac{\pi}{2} - \theta\). Thus, you should be able to remember, as definitions, \(\cos\left(\frac{\pi}{2} - x\right) = \sin(x), \sin\left(\frac{\pi}{2} - x\right) = \cos(x)\). The rest is just using symmetry properties to get it into a form like the above. Here are some examples.
 = \cos\left(-2\pi + \frac{\pi}{2} + x\right) = \cos\left(\frac{\pi}{2} - x\right) = \sin(x)<br />)
 = \sin\left(-\frac{\pi}{2} - x\right) = -\sin\left(\pi + \left(-\frac{\pi}{2} - x\right)\right) = -\sin\left(\frac{\pi}{2} - x\right)<br />)
<br />)
 = \sin\left(\pi - \left(\frac{\pi}{2} + x\right)\right) = \sin\left(\frac{\pi}{2} - x\right) = \cos(x)<br />)
That's all there is to it. You have to see whether you have to add or subtract 2\(\pi\), add or subtract \(\pi\), or subtract the angle from \(\pi\) or 2\(\pi\) to get it into the basic form.
AN LaTex is terrible.
Got no clue how to do iii. - probably missing something easy.
Any help is appreciated
Let O denote the origin and let A, B be the x, y axis intercepts of \(y = -2x + c\). If the midpoint M of AB is such that \(OM = 2\sqrt{5}\), find c.
Let's break this question down.
1. What does the question want? It wants c.
2. What does the question tell you? It tells you something about the midpoint of AB.
3. Does this information immediately look related to the question? Perhaps not, but let's work with it and see what happens.
4. What information is relevant to the midpoint? The midpoint is of AB, so perhaps we should find the coordinates of A and B to find where M is.
OK, now we've got this sorted, let's proceed. A is the x intercept, so you find this by setting \(y = 0\), which gives \(x = \frac{c}{2}\), so \(A = \left(\frac{c}{2},0\right)\). B is the y intercept, found by setting \(x = 0\), which gives \(y = c\) and thus \(B = \left(0,c\right)\).
Now that we've got A, B, we need to find M. The midpoint of a line segment has x, y coordinates which are the average of the x, y coordinates respectively of the endpoints. Thus, we find \(M = \{\frac{c}{4},\frac{c}{2}\).
Now that we have M, we can finally impose the requirement that the length \(OM = 2\sqrt{5}\). Note that O is the origin.
^2 + \left(\frac{c}{2}\right)^2} = |c|\sqrt{\frac{1}{16} + \frac{1}{4}} = \frac{|c|\sqrt{5}}{4} = 2\sqrt{5}\\<br />|c| = 8\\<br />c = \pm 8<br />)
where it is noted that if you want to take \(c^2\) out of the square root, \(\sqrt{c^2} = |c|\) to account for \(c\) being negative.
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