Memory Tip for Sine, Cosine, and Tangent of Special Angles (Trigonometry)

Trig Table


There is a simple way to remember the sine, cosine, and tangent of special trigonometry angles.

The special trig angles are 0º, 30º, 45º, 60º, and 90º. What makes these angles special? The 30º-60º-90º triangle is one-half of an equilateral triangle, while the 45º-45º-90º triangle is one-half of a square. In both cases, the trig functions (sine, cosine, and tangent) can be expressed as simple ratios.

Here is the trick for quickly working out the sine, cosine, and tangent of 0º, 30º, 45º, 60º, and 90º.

STEP 1: Special angles.

Write the special angles in order.

Trig Table top

STEP 2: Integers.

Write the integers 0 thru 4 in order.

Trig Table numbers

STEP 3: Squareroots.

Squareroot each number.

Trig Table roots

STEP 4: Find the sine of theta.

Divide each number by 2.

Trig Table sine

These are the sine of 0º, 30º, 45º, 60º, and 90º.

It’s that simple. Here’s a recap:

  • Write the numbers 0 thru 4.
  • Squareroot each number.
  • Divide each number by 2.

STEP 5: Find the cosine of theta.

Just write the previous numbers in reverse order.

Trig Table cos

Why does this work? Because the sine of theta equals the cosine of the complement of theta: sin(θ)=cos(90º–θ). What’s opposite to theta is adjacent to its complement.

STEP 6: Find the tangent of theta.

Divide sine theta by cosine theta.

Trig Table tan


This chart shows all of the steps together.

Trig Table Tips

  • Write the special angles.
  • Write the integers 0 thru 4.
  • Squareroot each number.
  • Divide each number by 2. This gives you sine of theta.
  • Write the numbers in reverse order. This gives you cosine of theta.
  • Divide the previous two rows (sine over cosine). This gives you tangent theta.


There are two different, yet equivalent ways, to write the above chart.

That’s because of the following properties of irrational numbers:


So, for example, there are alternative ways to express the following trig values:

Trig Values

The chart on this blog uses standard form. Most math courses use standard form, which means that there no irrational numbers (like root 2) in the denominator.

Sometimes you find the trig table in another nonstandard form. In that form, you see 1 over root 2 in place of root 2 over 2, and you see 1 over root 3 in place of root 3 over 3.

It’s important to realize that both forms are correct. The standard form, however, is expected in most math courses.


Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

  • Trigonometry Essentials Practice Workbook with Answers
  • Learn or Review Trigonometry Essential Skills
  • Trigonometry Flash Cards (for Kindle)
  • Algebra Essentials Practice Workbook with Answers
  • Systems of Equations: Simultaneous, Substitution, Cramer’s Rule
  • Other volumes cover fractions, long division, arithmetic, and more
  • Also look for books on the fourth dimension, astronomy, conceptual chemistry, and more

Multiplication Facts 0-10 Memory Tips

Multiplication Fives


When students learn multiplication, it starts out easy.

Too easy, as 0 times anything equals 0 (e.g. 0 x 8 = 0, while anything times 1 equals itself (e.g. 4 x 1 = 4).

But it doesn’t stay easy. Most students struggle to remember multiplication facts where one of the numbers is 6, 7, 8, or 9.

However, there are some tips to aid in memorization.

For one, mirror images don’t matter. The word for this is commutative.

This means 2 x 8 = 8 x 2, for example. Both equal 16.

So if you know 6 x 8 = 48, you also know that 8 x 6 = 48. You don’t need to memorize both.

One of the tricky multiplication facts is 7 x 8. But it’s easy if you remember the trick.

Remember 5678. These are 5 thru 8 in order. 56 = 7 x 8. Piece of cake, huh?

The 9’s are easy. The answer is one decade less than multiplying by 10, then make the two digits add up to 9.

For example, 9 x 7 is in the 60’s (because 10 x 7 = 70, one decade less is 60). It’s 63 since 6 + 3 = 9.

Another example is 5 x 9. It’s in the 40’s (one decade less than 5 x 10 = 50). It’s 45 since 4 + 5 = 9.

To do the trick, after you figure out the decade (by multiplying by 10 and then subtracting 10), subtract the tens digit from 9 to get the units digit.

For example, consider 9 x 6. Multiply 10 x 6 to get 60, and subtract 10 to make 50 (one decade less). Now subtract 5 (the tens digit of 50) from 9 to get 4. Therefore, 9 x 6 = 54.

Once you get the hang of it, this makes remembering the 9’s easy. Try out all the 9’s to get some practice.

If you’re good at doubling numbers quickly, try writing 6 as 2 x 3.

Then 6 x 7 = 2 x 3 x 7. If you know 3 x 7 = 21, double 21 to get 42.

Similarly, for 6 x 4 = 2 x 3 x 4, start with 3 x 4 = 12 and double 12 to make 24.

You can use the doubling trick for the 8’s, too. Just double the number 3 times.

For example, consider 5 x 8. Double 5 three times: 10, 20, 40. So 5 x 8 = 40.

Try 8 x 6. Double 6 three times: 12, 24, 48. Therefore, 8 x 6 = 48.

It works with the 4’s, also. Just double twice.

With 4 x 9, double 9 twice: 18, 36. So 4 x 9 = 36.

That leaves 7 x 7 = 49. You should know 7 x 9 from the 9’s trick.

You can make 7 x 6 and 7 x 8 from the doubling tricks. (The latter you can also know from the 5678 trick.)

5 and under are easier. So to complete the 7’s, you really just need to memorize 7 x 7 = 49.

This covers the 6 thru 9’s, which tend to be the trickier multiplication facts.

The 10’s and 11’s are easy. For the 10’s, just add a zero, as in 8 x 10 = 80 or 6 x 10 = 60.

For the 11’s, with 1 thru 9 just double the digits, like 3 x 11 =33 or 11 x 8 = 88. Get 11 x 10 = 110 from the 10’s trick (add a zero). Then you just need to memorize that 11 x 11 = 121 to complete the 11’s.

You can get the 12’s by doubling the 6’s. For example, knowing that 6 x 5 = 30, you can find that 12 x 5 = 60 by doubling 6 x 5.

Do you know any other tips for remembering multiplication facts 0-12? If so, please share them in the comments.


Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

  • arithmetic facts
  • multi-digit arithmetic
  • long division with remainders
  • fractions, decimals, and percents
  • algebra and trigonometry

Fibonacci Sequence & a Cool Pattern

Image from ShutterStock.

Image from ShutterStock.


The Fibonacci sequence adds consecutive terms:

  • 1
  • 1
  • 1 + 1 = 2
  • 2 + 1 = 3
  • 3 + 2 = 5
  • 5 + 3 = 8
  • 8 + 5 = 13
  • 13 + 8 = 21
  • 21 + 13 = 34
  • 34 + 21 = 55
  • 55 + 34 = 89

Since the last two terms were 55 and 89, we would add these together to get 89 + 55 = 144.

Then you would add 144 and 89 to make 233, and so on.

I saw a cool pattern involving the Fibonacci sequence recently at the Mathemagical Site:

Fibonacci Triples via Mathemagical Site

This involves Fibonacci triples.

A Fibonacci triple consists of three consecutive numbers from the Fibonacci sequence, such as:

  • 1, 1, 2
  • 1, 2, 3
  • 2, 3, 5
  • 3, 5, 8
  • 5, 8, 13
  • 8, 13, 21
  • 13, 21, 34

As shown on the Mathemagical Site, the square of the middle number is always one less or one more than the product of the first and third numbers:


Here are a few examples:

  • (2, 3, 5): 3 x 3 = 2 x 5 – 1
  • (3, 5, 8): 5 x 5 = 3 x 8 + 1
  • (5, 8, 13): 8 x 8 = 5 x 13 – 1

Curious about this, I’ve been working through the algebra, and finally came up with an algebraic proof, which follows.

My proof is algebraic and not necessarily obvious, but since I worked it out, I thought I would share it here. 🙂

We begin with two facts about the Fibonacci sequence:


These are two different ways of saying that if you add two consecutive numbers from the Fibonacci sequence, you get the next number in the sequence.

Now solve for xn in each sequence:


Multiply these together:


Foil this out:


Recall that


Plug this into the first term on the right-hand side of the previous equation:




Two of these terms cancel (the remaining terms are rearranged):


Believe it or not, this basically concludes the proof. The remainder is basically interpreting this result.

This is a recursion relation that relates the square of the n-th term to the square of the previous term (xn-1 times itself).

Following is the Fibonacci sequence, labeling values of n:

  • n = 1 is 1.
  • n = 2 is 1.
  • n = 3 is 2.
  • n = 4 is 3.
  • n = 5 is 5.
  • n = 6 is 8.
  • n = 7 is 13.
  • n = 8 is 21.

Let’s plug in n = 3 and see what happens:


If instead you plug in n = 4, you get:


Now just plug in these two expressions (x3x3 – x4x2 and x4x4 – x5x3) into the previous recursion relation and you can prove that all of the Fibonacci triples satisfy one of the following relations:


That is, if x3x3 – x4x2 = 1 and x4x4 – x5x3 = -1, the previous recursion relation gives similar expressions for x5x5 – x6x4, x6x6 – x7x5, and so on.


Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

  • Algebra Essentials Practice Workbook with Answers
  • Trigonometry Essentials Practice Workbook with Answers
  • Learn or Review Trigonometry: Essential Skills

Cursive Handwriting for Math Lovers

Cover designed by Melissa Stevens at



Julie Harper is the author of a popular series of handwriting workbooks.

I recently had the opportunity to coauthor a handwriting workbook with her, geared toward math lovers.

This unique book combines math terms and concepts with handwriting practice.

Find this book here:

Find Julie Harper’s handwriting books:

Find Chris McMullen’s math workbooks:


Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Algebra with Roman Numerals

Algebra Roman Numerals


Aren’t you glad you don’t have to do algebra with Roman numerals?

The problem is: Xx + X = CX.

This translates to 10x + 10 = 110.

Subtract X from both sides: Xx = CX – X = C.

Translation from Roman numerals: 10x = 110 – 10 = 100.

Now divide both sides by the coefficient: x = C/X = X.

Not expressed in Roman numerals, it looks like this: x = 100/10 = 10.


Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Practice Arithmetic with Geometric Dice



My daughter recently received a huge pack of cool geometric dice in several different colors:

  • An icosahedron (20-sided polyhedron) with the numbers 1 thru 20.
  • A dodecahedron (12-sided polyhedron) with the numbers 1 thru 12.
  • A decahedron (10-sided polyhedron) with the numbers 0 thru 9, and another with the numbers 0 thru 90.
  • An octahedron (8-sided polyhedron) with the numbers 1 thru 8.
  • A cube (6-sided polyhedron) with the numbers 1 thru 6.

These dice turned out to be really handy for learning addition and multiplication facts.


You can easily practice addition facts and multiplication facts with these dice.

Here are some examples:

  • ADDITION/MULTIPLICATION. Roll two decahedra, marked 0 thru 9. Add or multiply the two numbers to practice addition or multiplication facts 0 thru 9.
  • SMALLER NUMBERS. Roll two cubes, marked 1 thru 6. Add or multiply the two numbers to practice addition or multiplication facts 1 thru 6. The cubes let students focus on the smaller numbers first, before working with 7, 8, and 9. (If you want more basic practice, find tetrahedra—4-sided polyhedra—marked 1 thru 4.)
  • FOCUSED FACTS. Roll one decahedron, marked 0 thru 9. For example, suppose you want to practice your multiplication table of 4’s. Simply multiply the die by 4. This lets you concentrate on a single number’s addition or multiplication facts at a time.
  • 11 THRU 20. Advance to dodecahedra or icosahedra to practice the facts 1 thru 12 or 1 thru 20.
  • SUBTRACTION. Practice subtraction facts using an icosahedron and a decahedron. Be careful to subtract the smaller number from the larger number; sometimes, the number on the 10-sided die will be larger. (Advanced students who are learning about negative numbers can use these to sometimes subtract the larger number from the smaller number.)
  • TENS. Multiply powers of 10 using one decahedron with 0 thru 90 and another with 0 thru 9. Or roll one decahedron and multiply that by 10 for more basic tens practice.
  • POWERS. Roll a tetrahedron and a decahedron together to learn about powers. Let the tetrahedron serve as the exponent.
  • FRACTIONS. Roll four decahedra to learn about fractions. These will give you the numerators and denominators of two fractions. Then you can add them, multiply them, divide them, compare them (figure out which is bigger), or subtract them (but first find out which is larger).
  • DICE WAR. If you have several dice to divide equally, you can play dice war with a friend. Each player rolls two dice. Either add or multiply the numbers (choose one before the game begins). The higher sum or product collects both dice.


Another cool thing about using a variety of geometric dice to play math games is that kids get to hold various geometric solids in their hands, see how they look, get a feel for them, and after much use remember how many sides each shape has.

Better than just being told or shown what a dodecahedron is… hold one in your hands, roll it, play with it for months. Then you’ll ‘know’ that solid when you hear its name. (It helps when someone learns and uses the correct names while using the dice.)

Many of these dice packages are sold with role-playing games in mind, but there is no reason that you can’t use them for math practice instead.


  • Polyhedron: a three-dimensional solid.
  • Polygon: a two-dimensional object, not a solid; it’s flat.
  • Polyhedra is plural, polyhedron is singular.
  • Dice is plural, die is singular.


Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks