Trigonometry/Plotting (Cos t, Sin t)

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Plotting (Cos t, Sin t)

Exercise: Using Sine and Cosine on Your Calculator[edit]

It is nowadays very easy to find the value of sin and cos for any angle: many calculators have buttons that give these values, or on a PC you can use the calculator that comes with Windows. Spreadsheets can also calculate these functions.

Graph Paper
First find the \cos\, (or cosine) button on your calculator. Type in 60 and press \cos\,. You should see 0.5 on the screen.

Any number of things could have gone wrong here. You might not have \cos\,, you might not have a calculator, you might not get the answer 0.5. Here's what to do if it didn't work.

  • If you don't have a calculator or your calculator does not have \cos\, then visit this page which explains How to use the Windows Scientific Calculator.
  • If you didn't get the answer 0.5 but instead got -0.9524... or an error, then go to the page on how to use the Windows Scientific Calculator and read the last bit about changing between degrees, radians and gradians. You may be able to use the windows calculator, or from the description on that page see what to do to get your calculator using degrees.

OK. So now type in 60 and this time press \sin\, (or sine). This time you should get the answer 0.8660... If this is working fine, we'd like you to draw up a table for coordinates. We've filled in some of the values, but you need to fill in the rest.

\displaystyle t \displaystyle \cos t \displaystyle \sin t
\displaystyle 0^\circ \displaystyle 1.00 \displaystyle 0.00
\displaystyle 70^\circ \displaystyle 0.94
\displaystyle 110^\circ \displaystyle -0.34
\displaystyle 45^\circ
\displaystyle 60^\circ \displaystyle 0.50
\displaystyle -60^\circ \displaystyle 0.50
 
 
 

Whilst you are doing this you might like to check that the values we already filled in are correct. Did we get any wrong? The three rows at the end of the table are for you to choose values of t and then put in cosine and sine values for those values of t. We suggest you chose values for t that are somewhere between -900 and +900.

The piece of graph paper on this page is a clue as to what to do next. We'd like you to treat each of the pairs ( cos t, sin t) as a coordinate of a point on the graph paper and plot those points. Don't join them up yet, just plot the points. All the values you got should be between 1 and -1, so the axes already drawn on the graph should be the right ones. Oh, no one has drawn them yet. For now you will have to do that part too. This book is still being written. Anyway, once you start plotting some of the points you should start to get an idea of what shape is being formed. Even if you know the answer already or can guess it, it can be worth doing this exercise. It gives you a really good sense of what cosine and sine are.

Chances are the shape you have drawn so far is very incomplete. You should be able to choose some values for t that will help to make it more complete.


Something more challenging...

If you are finding this all ridiculously easy, try and find another values of t that give you the same answer as you got for \displaystyle 45^\circ. You should be able to find at least one.

  • Hint: try starting from t=444.
    • Should you increase t or decrease it to go in the right direction?