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Sines and Cosines of Common Angles

The Brain Dump

When my kids were going to high school, they found it very difficult to remember such things as the sine of 225�, or the cosine of 120�.  And then when it came time to take a test, they got flustered, because it required spitting out so darn many of these sines and cosines.

So I told them the "Brain Dump" trick, and it worked wonders.  Maybe it will help you, too.

First, let's take a look at a graph of the cosine function:

The Cosine Function

There are only a handful of special values to remember: 1/2, sqrt(2)/2, sqrt(3)/2, 1, and their negatives, and zero.  The cosine starts out at 1, passes neatly through each of these special values -- first the positives, then the negatives -- then back up again, hitting each value in reverse order.

Can you remember this picture?

Good, now let's look at the sine function:

The Sine Function

This may be the more familiar curve, often called the "sine wave" because of the way it resembles a wave on the surface of a body of water.  The sine starts at zero, goes up through each special value to 1, then hits the same values, then their negatives, reaching -1, and then back to zero to finish.

Now, let's make the "Brain Dump" table of cosines and sines.  It looks like this:

DegreesRadians cossintan
001 00
30π/6 sqrt(3)/21/2sqrt(3)/3
45π/4 sqrt(2)/2sqrt(2)/21
60π/31/2 sqrt(3)/2sqrt(3)
90π/20 1undefined
1202π/3 -1/2sqrt(3)/2-sqrt(3)
1353π/4 -sqrt(2)/2sqrt(2)/2-1
1505π/6 -sqrt(3)/21/2-sqrt(3)/3
180π-1 00
2107π/6 -sqrt(3)/2-1/2sqrt(3)/3
2255π/4 -sqrt(2)/2-sqrt(2)/21
2404π/3 -1/2-sqrt(3)/2sqrt(3)
2703π/20 -1undefined
3005π/3 1/2-sqrt(3)/2-sqrt(3)
3157π/4 sqrt(2)/2-sqrt(2)/2-1
33011π/6 sqrt(3)/2-1/2-sqrt(3)/3
3601 00

I want you to make your own copy of this table, and you should be able to create such a table at will from memory.  That's why I call it your Brain Dump.  If you remember the special angles -- all the multiples of 30�, and all the multiples of 45� -- then the first thing you should not have any trouble with this.  If you need to know radians, then include them in your table.

Memory Trick

Do you have trouble remembering
the special values of cosine and
sine?  They are...

sqrt(3)/2, and

Here's how to remember them:

sqrt(0)/2 = 0,
sqrt(1)/2 = 1/2,
sqrt(2)/2 = sqrt(2)/2,
sqrt(3)/2 = sqrt(3)/2, and
sqrt(4)/2 = 1.

So, you see, the special values are
nothing more than the square roots
of zero through four, each divided
by two.  Does this help?  If not,
forget about it!  Pick and choose
what helps you.

Here's how to fill in the other columns.  The cosine starts at 1, remember?  Then it hits sqrt(3)/2, sqrt(2)/2, 1/2, and 0.  Just write these in your table, and make sure the 1's and 0's are all at multiples of 90�.  Then the cosine keeps on dipping into negative territory (look at the graph, above).  So write the negatives of the special values, but in reverse order.  When you get to negative 1, which should be at 180�, then you write all the special values again, in completely reverse order.  So, you see, making a table like this involves very little actual thinking or remembering, and a lot of copying.

Now, to make the sine column, you can either remember that sine starts at 0, goes up to 1, then down to -1, and then back up to zero, always cycling through each of the special values.


you can put your pencil at the 90� row of the sine column, and copy the cosine column.  When you get to the bottom of your table, you can "wrap" back to the top, remembering to start the sine column with a zero.

For the tan column, if you need one, you can just divide sine by cosine.  Then you have to remember that 1/sqrt(3) is the same as sqrt(3)/3.  Or, you can remember these special values for tan, which are different from the ones for sine and cosine -- it's up to you.

Making a Table for your Trig Test

What good is a lovely table like this when you need to take a test, right?  I hear you.  Well, you wouldn't dare bring this table to your classroom, and try to sneak a peek at it during your test, would you?  That would be cheating.

So here's what you can do, and it's perfectly legal:

Time yourself making this table on a blank piece of paper.  Use only the columns that you will need for your test.  It will probably take you fifteen minutes the first time you do it, and you'll have to look at this web page once or twice along the way.  But the second time, you will probably be able to do it in 10 minutes.  Then 5.  My kids learned to write out a table like this in under three minutes.  How long did that take to learn how to write a table like this in three minutes?  Did it take you an hour to practice?  It's the best hour you ever spent studying for your trig test, I promise.

Now, when your sitting in your classroom, ready to take the test, the first thing you should do (or maybe the second thing, right after you read the questions on the test) is take a piece of scratch paper, and write out this table.  Don't share it with anyone during the test -- that would be cheating.  But it's not cheating to write anything you need to use on your scratch paper, as long as you do it from memory!  After you write it, check it carefully, to make sure all the signs are right -- apart from the zeros, exactly half of the signs should be minus, and the other half should be plus.  Make sure the cosine starts high, dips into the negative numbers, then ends high.  Make sure the sine starts at zero, goes up to 1, then all the way down to -1, then back to zero.

It should take you three or four minutes to write out the table, then another minute or two to check it.  Five minutes well spent during your short test period, because then the questions will be easy: just look up the answers on the table!


Internet references, which cites Ptolemy's On the Size of Chords Inscribed in a Circle (2nd Century AD).

Related Pages in this website

Trig Equivalences

Special Angles -- sines and cosines of additional angles, such as 15� and 54�

Trig functions of special angles, part 2 -- cos 40� is one of the roots of 8x3-6x+1, so can we find an arithmetic expression for cos 40�?

Sin or Cos 3x, 4x, etc. -- trig functions of any multiple of an angle.

Golden Ratio -- (sqrt(5)+1)/2, a special number that comes up in a variety of geometrical contexts



The webmaster and author of this Math Help site is Graeme McRae.