Tag Archives: triangles

SOHCAHTOA

SOHCAHTOA (pronounced ‘soccer-toe-uh’) is a useful ‘mnemonic’ to remember the definitions of sines, cosines and tangents. Amazingly, I was never taught this at school, so I just had to look up all the funny numbers in a big book of tables without understanding what they meant. As a result, I was always a bit confused by trigonometry until I started teaching Maths and came across SOHCAHTOA quite by accident!

The reason it’s called SOHCAHTOA is because the letters of all three equations make up that word – if you ignore the equals signs…

First of all, let’s define our terms:

  • S stands for sine (or sin)
  • O stands for the opposite side of a right-angled triangle
  • H stands for the hypotenuse of a right-angled triangle
  • C stands for cosine (or cos)
  • A stands for the adjacent side of a right-angled triangle
  • T stands for tangent (or tan)
  • O stands for the opposite side of a right-angled triangle (again)
  • A stands for the adjacent side of a right-angled triangle (again)

Sines, cosines and tangents are just the numbers you get when you divide one particular side of a right-angled triangle by another. For a given angle, they never change – however big the triangle is.

Sine = Opposite ÷ Hypotenuse

Cosine = Adjacent ÷ Hypotenuse

Tangent = Opposite ÷ Adjacent

All these ratios were discovered by Indian and Arabic mathematicians some time before the 9th Century, but you can still use them today to help you work out the length of the sides of a right-angled triangle or one of the angles.

Each of these formulas can be rearranged to make two other formulas. (If it helps, you can put the three values in a number triangle with the one in the middle at the top). Let’s take the sine formula first:

Sine = Opposite ÷ Hypotenuse means:

  • Hypotenuse = Opposite ÷ Sine
  • Opposite = Hypotenuse x Sine

As long as you know the angle and the length of the opposite side or the hypotenuse, you can work out the length of the other one of those two sides.

  • Unknown: hypotenuse
    Known: opposite and angle
    • If one of the angles of a right-angled triangle is 45° and the opposite side is 5cm, the formula for the length of the hypotenuse must be opposite ÷ sin(45°). The sine of 45° is 0.707 (to three decimal places), so hypotenuse = 5 ÷ 0.707 = 7cm (to the nearest cm).
  • Unknown: opposite
    Known: hypotenuse and angle
    • If one of the angles of a right-angled triangle is 45° and the hypotenuse is 5cm, the formula for the length of the opposite side must be hypotenuse x sin(45°). The sine of 45° is 0.707 (to three decimal places), so opposite = 5 x 0.707 = 4cm (to the nearest cm).

Equally, as long as you know the the hypotenuse and opposite side lengths, you can work out the angle by using the ‘arcsine’ or ‘inverse sine’ function on your calculator, which works out the matching angle for a given sine and is written as sin-1, eg sin(45°) = 0.707, which means sin-1(0.707) = 45°.

  • Unknown: angle
  • Known: opposite and hypotenuse
    • If the opposite side of a right-angled triangle is 4cm and the hypotenuse is 5cm, the formula for the angle must be sin-1(4÷5), or the inverse sine of 0.8. The sine of 53° (to the nearest degree) is 0.8, so the angle must be 53°.

We can do the same kind of thing with the cosine formula, except this time we’re dealing with the adjacent rather than the opposite side.

Cosine = Adjacent ÷ Hypotenuse means:

  • Hypotenuse = Adjacent ÷ Cosine
  • Adjacent = Hypotenuse x Cosine

As long as you know the angle and the length of the adjacent side or the hypotenuse, you can work out the length of the other one of those two sides.

  • Unknown: hypotenuse
    Known: adjacent and angle
    • If one of the angles of a right-angled triangle is 45° and the adjacent side is 5cm, the formula for the length of the hypotenuse must be adjacent ÷ cos(45°). The cosine of 45° is 0.707 (to three decimal places), so hypotenuse = 5 ÷ 0.707 = 7cm (to the nearest cm).
  • Unknown: adjacent
    Known: hypotenuse and angle
    • If one of the angles of a right-angled triangle is 45° and the hypotenuse is 5cm, the formula for the length of the adjacent side must be hypotenuse x cos(45°). The sine of 45° is 0.707 (to three decimal places), so adjacent = 5 x 0.707 = 4cm (to the nearest cm).

Equally, as long as you know the the hypotenuse and adjacent side lengths, you can work out the angle by using the ‘arccosine’ or ‘inverse cosine’ function on your calculator, which works out the matching angle for a given cosine and is written as cos-1, eg cos(45°) = 0.707, which means cos-1(0.707) = 45°.

  • Unknown: angle
  • Known: adjacent and hypotenuse
    • If the adjacent side of a right-angled triangle is 4cm and the hypotenuse is 5cm, the formula for the angle must be cos-1(4÷5), or the inverse cosine of 0.8. The sine of 37° (to the nearest degree) is 0.8, so the angle must be 37°.

Finally, we can do the same kind of thing with the tangent formula, except this time we’re dealing with the opposite and adjacent sides.

Tangent = Opposite ÷ Adjacent means:

  • Adjacent = Opposite ÷ Tangent
  • Opposite = Adjacent x Tangent

As long as you know the angle and the length of the opposite or adjacent side, you can work out the length of the other one of those two sides.

  • Unknown: adjacent
    Known: opposite and angle
    • If one of the angles of a right-angled triangle is 45° and the opposite side is 5cm, the formula for the length of the adjacent must be opposite ÷ tan(45°). The tangent of 45° is 1, so adjacent = 5 ÷ 1 = 5cm.
  • Unknown: opposite
    Known: adjacent and angle
    • If one of the angles of a right-angled triangle is 45° and the adjacent side is 5cm, the formula for the length of the opposite side must be adjacent x tan(45°). The tangent of 45° is 1, so opposite = 5 x 1 = 5cm.

Equally, as long as you know the the opposite and adjacent side lengths, you can work out the angle by using the ‘arctangent’ or ‘inverse tangent’ function on your calculator, which works out the matching angle for a given tangent and is written as tan-1, eg tan(45°) = 0.707, which means tan-1(0.707) = 45°.

  • Unknown: angle
  • Known: adjacent and hypotenuse
    • If the adjacent side of a right-angled triangle is 5cm and the hypotenuse is 5cm, the formula for the angle must be tan-1(5÷5), or the inverse tangent of 1. The tangent of 45° is 1, so the angle must be 45°.

Useful Formulas

Useful formulas

Which one of these is it again…?

What is a problem? A problem = a fact + a judgment. That is a simple formula that tells us something about the way the world works. Maths is full of formulas, and that can intimidate some people if they don’t understand them or can’t remember the right one to use.

However, formulas should be our friends, as they help us to do sometimes complex calculations accurately and repeatably in a consistent and straightforward way. The following is a list of the most useful ones I’ve come across while teaching Maths to a variety of students at a variety of ages and at a variety of stages in their education.

Averages

  • The mean is found by adding up all the values and dividing the total by how many there are, eg the mean of the numbers 1-10 is 5.5, as 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55, and 55 ÷ 10 = 5.5.
  • The mode is the most common value (or values), eg the mode of 1, 2, 2, 3, 4, 5 is 2.
  • The median of an odd number of values sorted by size is the one in the middle, eg the median of the numbers 1-5 is 3. The median of an even number of values is the mean of the two numbers in the middle, eg the median of the numbers 1-10 is 5.5, as 5 and 6 are the numbers in the middle, and 11 ÷ 2 = 5.5.
  • The range is the highest value minus the lowest, eg the range of the numbers 1-10 = 10 – 1 = 9.

Geometry

  • Angles around a point add up to 360º
  • Angles on a straight line add up to 180º
  • Opposite angles are equal, ie the two pairs of angles opposite each other when two straight lines bisect (or cross) each other
  • Alternate angles are equal, ie the angles under the arms of a ‘Z’ formed by a line (or ‘transversal’) bisecting two parallel lines
  • Corresponding angles are equal, ie the angles under the arms of an ‘F’ formed by a line (or ‘transversal’) bisecting two parallel lines
  • Complementary angles add up to 90º
  • Any straight line can be drawn using y = mx + c, where m is the gradient and c is the point where the line crosses the y-axis (the ‘y-intercept’)
  • The gradient of a straight line is shown by δy/δx (ie the difference in the y-values divided by the difference in the x-values of any two points on the line, usually found by drawing a triangle underneath it)

Polygons

  • Number of diagonals in a polygon = (n-3)(n÷2) where n is the number of sides
  • The sum of the internal angles of a polygon = (n-2)180º, where n is the number of sides
  • Any internal angle of a regular polygon = (n-2)180º ÷ n, where n is the number of sides

Rectangles

  • Perimeter of a rectangle = 2(l + w), where l = length and w = width
    Note that this is the same formula for the perimeter of an L-shape, too.
  • Area of a rectangle = lw, where l = length and w = width

Trapeziums

  • Area of a trapezium = lw, where l = average length and w = width (in other words, you have to add both lengths together and divide by two in order to find the average length)

Triangles (Trigonometry)

  • Area of a triangle = ½bh, where b = base and h = height
  • Angles in a triangle add up to 180º
  • Pythagoras’s Theorem: in a right-angled triangle, a² + b² = c², ie the area of a square on the hypotenuse (or longest side) is equal to the sum of the areas of squares on the other two sides

 

 

 

 

 

 

Circles

  • Circumference of a circle = 2πr, where r = radius
  • Area of a circle = πr², where r = radius
  • π = 3.14 to two decimal places and is sometimes given as 22/7

Spheres

  • Volume of a sphere = 4/3πr³, where r = radius
  • Surface area of a sphere = 4πr², where r = radius

Cuboids

  • Volume of a cuboid = lwh, where l is length, w is width and h is height
  • Surface area of a cuboid = 2(lw + lh + wh), where l is length, w is width and h is height

Number Sequences

  • An arithmetic sequence (with regular intervals) = xn ± k, where x is the interval (or difference) between the values, n is the value’s place in the sequence and k is a constant that is added or subtracted to make sure the sequence starts at the right number, eg the formula for 5, 8, 11, 14…etc is 3n + 2
  • The sum of n consecutive numbers is n(n + 1)/2, eg the sum of the numbers 1-10 is 10(10 + 1)/2 = 110/2 = 55

Other

  • Speed = distance ÷ time
  • Profit = sales – cost of goods sold
  • Profit margin = profit ÷ sales
  • Mark-up = profit ÷ cost of goods sold