1. ## Trigonometry help, relatively basic, wanting to check answers.

Hello everybody,

I am just wanting to check my answers to the following questions, and if i am wrong could somebody please point out how to do it correctly?

A satellite orbiting the Earth is rotating twice a minute. As the satellite turns
an instrument on the side of the satellite is periodically heated the by the sun
then cools as it moves into the shady side of the satellite. The instrument
reaches a maximum temperature of 50◦C and a minimum temperature of -
10◦C. Supposing that t = 0 corresponds to the time the temperature of the
instrument is at its highest write down a trigonometric function that captures
this behaviour. Sketch the temperature of the instrument over one minute (that
is two revolutions of the satellite).

Function = Function: f(t) = 30cos2t Is that correct?

The volume of water in a tank at a particular time (measured in seconds) is
given by V (t) = 2t(1 - t)m3. Find the rate of change of the volume of water
in the tank from first principles (i.e. using the definition of the rate of change -
which is a limit).

I got -2m^3/s.

Consider the function f(x) = -x2+3x-1. Find the equation of the tangent to
the graph of f(x) at x = 2. [NOTE: when calculating f′(2), use first principles.]

f(x) = -x^2+3x-1
f(2) = 1
f '(x) = -2x+3
f '(2) = -1
tangent line: y = -1x + b -- substitute (2,1) into equation to solve for b
y = -x + 3

Consider the trigonometric function f(t) = 4 - 2 cos(2piet).
What is the amplitude of f(t)?
What is the period of f(t)?
What are the maximum and minimum values attained by f(t)?
Sketch the graph of f(t) for t 2 [0; 2].

Can somebody help me out with this one please?

Thank you in advance for any help .

2. ## Re: Trigonometry help, relatively basic, wanting to check answers.

You are correct that a cosine function models the situation the easiest, but your function is incorrect.

First of all, the mean temperature would be \displaystyle \begin{align*} 30^{\circ} \end{align*}, which has a difference of \displaystyle \begin{align*} 20^{\circ} \end{align*} from the maximum or minimum. So the amplitude is \displaystyle \begin{align*} 20 \end{align*} and the constant added to the function is \displaystyle \begin{align*} 30 \end{align*}.

I assume t is measured in seconds. You know that it rotates twice a minute, so once every 30 seconds. This is the period. But in your function, the b value is given by

\displaystyle \begin{align*} \textrm{Period}&= \frac{2\pi}{b} \\ 30 &= \frac{2\pi}{b} \\ 30b &= 2\pi \\ b &= \frac{\pi}{15} \end{align*}

So your function should be \displaystyle \begin{align*} f(t) = 20\cos{\left(\frac{\pi t}{15}\right)} + 30 \end{align*}.

3. ## Re: Trigonometry help, relatively basic, wanting to check answers.

Originally Posted by lkt004
The volume of water in a tank at a particular time (measured in seconds) is
given by V (t) = 2t(1 - t)m3. Find the rate of change of the volume of water
in the tank from first principles (i.e. using the definition of the rate of change -
which is a limit).

I got -2m^3/s.
This is obviously incorrect. Checking your answer using simple differentiation rules should tell you that...

\displaystyle \begin{align*} \frac{d}{dt}\left[2t(1 - t)\right] &= \frac{d}{dt}\left[2t - 2t^2\right] \\ &= 2 - 4t \end{align*}

Anyway, you need to use first principles. So since \displaystyle \begin{align*} f(t) = 2t(1 - t) = 2t - 2t^2 \end{align*} you need to evaluate

\displaystyle \begin{align*} f'(t) = \lim_{h \to 0}\frac{f(t + h) - f(t)}{h} \end{align*}

4. ## Re: Trigonometry help, relatively basic, wanting to check answers.

Thank you, i am just doing some quick revision before i tackle discrete maths as an elective (just for fun!) and it seems i have alot of work to do! Could you elaborate on the evaluation of the final equation a touch? Not sure where to start on that one.

Consider the trigonometric function f(t) = 4 - 2 cos(2piet).
What is the amplitude of f(t)?
What is the period of f(t)?
What are the maximum and minimum values attained by f(t)?

Could you help out with this for me? I have my answers and working somewhere, will see if i can find it!

Thanks again, help is greatly appreciated (have not done maths since final year of high school!)

5. ## Re: Trigonometry help, relatively basic, wanting to check answers.

Originally Posted by lkt004
Thank you, i am just doing some quick revision before i tackle discrete maths as an elective (just for fun!) and it seems i have alot of work to do! Could you elaborate on the evaluation of the final equation a touch? Not sure where to start on that one.

Consider the trigonometric function f(t) = 4 - 2 cos(2piet).
What is the amplitude of f(t)?
What is the period of f(t)?
What are the maximum and minimum values attained by f(t)?
you need to review the basics of trig function (sine & cosine) graphs. have a look at this lesson ...

Trig/Graphs.pdf