Hello Drdj Originally Posted by
Drdj Dear all,
Could you kindly help me with these 2 sums - Q 18 and Q21 - many, many thanks
I think one of the reasons no-one has attempted to answer question 18 is that the diagram is not clear. We can't see the graph of $\displaystyle y = 4\cos3x$ at all. This means that it's not possible to be sure what the value of $\displaystyle k$ is. However, here is some working and a diagram showing two possible solutions. You will have to take it from here.
First $\displaystyle y = 4\cos3x$ crosses the $\displaystyle x$-axis where: $\displaystyle \cos3x = 0$
So, working in degrees:$\displaystyle \Rightarrow 3x = 90, 270, 450, 630, ...$
$\displaystyle \Rightarrow x = 30, 90, 150, 210, ...$
See the black graph in the attached diagram.
So let's suppose first that this graph meets $\displaystyle y= 2\sin x + k$ where $\displaystyle x = 30^o$. Then: $\displaystyle 2\sin30+k=0$
$\displaystyle \Rightarrow k = -1$
So that's one possible value of $\displaystyle k$. See the yellow coloured graph.
But perhaps the diagram shows a different position of the graph, and they actually meet at a different point at which the first graph crosses the $\displaystyle x$-axis; for instance, where $\displaystyle x = 210^o$. Then we get: $\displaystyle 2\sin210+k = 0$
$\displaystyle \Rightarrow k = 1$
This is the pink graph.
Over to you. Can you sort it out now?
For question 21: multiply top-and-bottom by $\displaystyle (1-\sin x)$: $\displaystyle \sqrt{\frac{1-\sin x}{1+\sin x}}= \sqrt{\frac{(1-\sin x)^2}{1-\sin^2 x}}$$\displaystyle = \sqrt{\frac{(1-\sin x)^2}{\cos^2 x}}$
$\displaystyle =\frac{1-\sin x}{\cos x}$, if we are given that $\displaystyle x$ is acute, because $\displaystyle \cos x$ is then positive.
$\displaystyle =\sec x - \tan x$
But if $\displaystyle x$ is obtuse, $\displaystyle \sin x > 0$ and $\displaystyle \cos x < 0$. So $\displaystyle \cos x = -\sqrt{\cos^2x}$. So we would have to say:$\displaystyle \sqrt{\frac{(1-\sin x)^2}{\cos^2 x}}=-\frac{1-\sin x}{\cos x}$$\displaystyle =\tan x - \sec x$
Can you see what happens for other values of $\displaystyle x$? If you're not sure, look at the graphs of $\displaystyle \sec x$ and $\displaystyle \tan x$, and see whereabouts $\displaystyle \sec x >\tan x$ and whereabouts $\displaystyle \tan x > \sec x$. And then bear in mind that $\displaystyle \sqrt{\frac{1-\sin x}{1+\sin x}}$ has to be positive.
Grandad