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SOHCAHTOA

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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 a side in 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.

  • 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.

  • 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.

  • 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 side 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°.

Sample Questions

  1. If one of the angles of a right-angled triangle is 20° and the opposite side is 15cm, how long is the hypotenuse?
  2. If one of the angles of a right-angled triangle is 35° and the hypotenuse is 7cm, how long is the opposite side?
  3. If the opposite side of a right-angled triangle is 3cm and the hypotenuse is 8cm, what is the angle?
  4. If one of the angles of a right-angled triangle is 75° and the adjacent side is 12cm, how long is the hypotenuse?
  5. If one of the angles of a right-angled triangle is 45° and the hypotenuse is 5cm, how long is the adjacent side?
  6. If the adjacent side of a right-angled triangle is 15cm and the hypotenuse is 2cm, what is the angle?
  7. If one of the angles of a right-angled triangle is 15° and the opposite side is 20cm, how long is the adjacent side?
  8. If one of the angles of a right-angled triangle is 45° and the adjacent side is 5cm, how long is the opposite side?
  9. If the adjacent side of a right-angled triangle is 15cm and the hypotenuse is 12cm, what is the angle?
  10. If the adjacent side of a right-angled triangle is 4cm and the hypotenuse is 18cm, what is the angle?

     

     

     

     

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Useful Formulas

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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 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.

Sample Questions

  1. What is the mean of the numbers 6, 5, 8, 2 and 4?
  2. What is the mode of the numbers 1, 5, 5, 12 and 3?
  3. What is the median of the numbers 2, 8, 9, 6 and 5?
  4. What is the median of the numbers 1, 13, 4, 6, 8 and 20?
  5. What is the range of the numbers 15, 2, 3, 8 and 4?

Circle Theorems

  • Alternate segment theorem
    The angle between a tangent and a chord is equal to the angle subtended by the same chord in the alternate segment.
  • Angles at the centre and at the circumference
    The angle formed at the centre of the circle by lines starting from two points on the circumference is double the angle formed on the other side of the circle by lines originating from the same points.
  • Angles in the same segment
    If you create two triangles between the ends of the chord and the side of the circle, the angles in the same segment are equal.
  • Angles in a semicircle
    If you draw a triangle using the diameter and any point on the circumference, the angle at that point is 90°.
  • Perpendicular chord bisection
    The perpendicular from the centre of a circle to a chord will always bisect the chord (divide it into two equal parts).
  • Tangent of a circle
    a) The angle between a tangent and radius is 90°.
    b) Tangents that meet at the same point outside the circle are equal in length.
  • Cyclic quadrilateral
    The opposite angles in a cyclic quadrilateral add up to 180°.
  • Angles subtended on the same arc
    Angles formed from two points on the circumference of a circle are equal to other angles, in the same arc, formed from those two points.
  • Angle in a semi-circle
    Angles created by drawing lines from the ends of the diameter of a circle to the circumference form a right angle.

Geometry

  • Angles around a point add up to 360º
  • Angles on a straight line add up to 180º
  • Vertically 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

Sample Questions

  1. How many diagonals are there in a square?
  2. How many diagonals are there in an octagon?
  3. What is the sum of the internal angles of a pentagon?
  4. What is the sum of the internal angles of a hexagon?
  5. What are the internal angles of a regular decagon?
  6. What are the internal angles of a regular heptagon?

Probability

  • The probability of something happening = the number of ways of getting what you want divided by the number of possible outcomes, eg the chance of getting heads if you flip a coin is 1/2 (or 0.5 or 50%).
  • The probability of a sequence of events = the probability of all the events multiplied together, eg the chance of getting two heads in a row if you flip a coin is 1/2 x 1/2 = 1/4 (or 0.25 or 25%).

Sample Questions

  1. What is the probability of rolling a six?
  2. If there are two red and three black balls in a bag, what is the probability of taking out a red one?
  3. What is the probability of rolling a number greater than 3?
  4. What is the probability of tossing three heads in a row?
  5. If the chance of hitting treble 20 is 1/10, what is the probability of scoring 180 with three darts?

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

Sample Questions

  1. What is the perimeter of a rectangle measuring 5 x 7 cm?
  2. What is the perimeter of a rectangle measuring 13 x 4 cm?
  3. What is the perimeter of an L-shape measuring 12 x 8 m overall with a 4 x 2 m piece missing?
  4. What is the area of a rectangle measuring 3 x 12 cm?
  5. What is the area of a rectangle measuring 6 x 8 cm?

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 to find the average length)

Sample Questions

  1. What is the area of a trapezium with a height of 7 cm and horizontal sides of 4 and 8 cm?
  2. What is the area of a trapezium with a height of 10 inches and horizontal sides of 5 and 4 inches?
  3. What is the area of a trapezium with a height of 5 cm and horizontal sides of 2 and 4 cm?
  4. What is the area of a trapezium with a height of 12 m and horizontal sides of 6 and 9 m?
  5. What is the area of a trapezium with a height of 10 cm and horizontal sides of 4 and 5 cm?

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)

Sample Questions

  1. What is the area of a triangle with a height of 5 cm and a base of 3 cm?
  2. What is the remaining internal angle of a triangle if the others are 20º and 30º?
  3. What is the length of the remaining side of a right-angled triangle if the others are 4 cm and 3 cm?
  4. What is the circumference of a circle with a radius of 5cm, assuming π = 3.14?
  5. What is the area of a circle with a diameter of 4 m, assuming π = 22/7?

Spheres

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

Sample Questions

  1. What is the surface area of a sphere with a radius of 15 cm, assuming π = 3.14?
  2. What is the surface area of a sphere with a diameter of 7 cm, assuming π = 3.14?
  3. What is the volume of a sphere with a radius of 1 cm, assuming π = 22/7?
  4. What is the volume of a sphere with a radius of 5 cm, assuming π = 3.14?
  5. What is the volume of a sphere with a diameter of 5 m, assuming π = 22/7?

Cuboids

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

Sample Questions

  1. What is the surface area of a cuboid measuring 3 x 4 x 5 cm?
  2. What is the surface area of a cuboid measuring 6 x 2 x 12 cm?
  3. What is the volume of a cuboid measuring 8 x 5 x 6 cm
  4. What is the volume of a cuboid measuring 2 x 15 x 7 cm
  5. What is the volume of a cuboid measuring 18 x 9 x 2 cm

Number Sequences

  • The nth term of any 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
    • If you know where the term is in the sequence, you can find it by plugging in n and solving the equation, eg the fourth term of the sequence above = 3n + 2 = 3 x 4 + 2 = 14
    • If you know what the term is, you can find its position in the sequence by adding the number as the result of the formula and solving for n, eg if the term in the sequence above = 14, then 3n + 2 = 14, 3n = 12 and n = 4.
  • 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

Sample Questions

  1. What is the formula for the nth term of 2, 4, 6, 8…?
  2. What is the formula for the nth term of -5, -3, -1, 1…?
  3. What is the seventh term of the sequence 4, 8, 12, 16…?
  4. What position is 66 in the sequence 3, 6, 9, 12…?
  5. What is the sum of the numbers 12 to 18?

Other

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

Sample Questions

  1. What is the speed of a vehicle that travels 20 miles in two hours?
  2. How long does it take to travel 210 km at 70 km/h?
  3. What profit does a butcher make on three steaks he buys at £2 and sells for £2.50?
  4. What’s his profit margin on those steaks?
  5. What’s his mark-up on those steaks? 

     

     

     

    If you’re looking for past papers with answers, especially in the run-up to 11+/13+ exams, GCSEs or A-levels, you can visit my Past Papers page and subscribe for just £37.99 a year.