Maths

Trigonometry - Higher

Trigonometry can be used to calculate the lengths of sides and sizes of angles in right-angled triangles.

The three formulae: sin, cos, tan

The sides of the right-angled triangles are given special names - the hypotenuse, the opposite and the adjacent.

The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides relate to the angle under consideration.

Click on the correct side of the triangle on the diagram below.

There are three formulae involved in trigonometry:

sin = opposite / hypotenuse

cos = adjacent / hypotenuse

tan = opposite / adjacent

Which formula you use will depend on the information given in the question.

There are a couple of ways to help you remember which formula to use. Remember SOHCAHTOA (it sounds like 'Sockatoa') or Some Old Hag Cracked All Her Teeth On Apples.

Finding the length of the opposite side

Question

Find the length of side BC.

image: 7cm traingle

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Answer
  • We are given angle A and side AB.
  • AB is the hypotenuse.
  • BC is opposite angle A.
  • Therefore we use the formula:
  • sin = opposite / hypotenuse
  • sin 30 = BC / 7
  • Multiply both sides by 7
  • 7 x sin30 = BC
  • Using a calculator we get
  • BC = 3.5cm

Now you try this one.

Question

Find length of side PQ, giving your answer correct to 3 sf.

image: triangle

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Answer

The answer is 4.77cm (3 sf).

  • Can you see that PQ is opposite the given angle and that QR is adjacent to it?
  • Use the formula:
  • tan = opposite / adjacent
  • tan50 = PQ / 4
  • 4 × tan50 = PQ
  • PQ = 4.77 (3 sf)

Re-arranging the formula

Question

Find YZ, giving your answer correct to 3 sf

triangle with 25 degree angle

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Answer
  • We are given angle Z
  • XZ is adjacent to angle Z.
  • YZ is the hypotenuse.
  • Therefore we use the formula: cos = adjacent / hypotenuse
  • cos 25 = 5 over YZ

  • This time our unknown side (YZ) is the denominator. If we multiply both sides by YZ, we get:
  • YZ x cos 25 = 5 over YZ x YZ

  • YZ x cos25 = 5
  • Now we can divide both sides by cos25:

    YZ x cos 25 over cos 25 = 5 over 25

  • YZ = 5 over cos 25

  • Using a calculator:
  • YZ = 5 ÷ 0.9063 = 5.52cm (3 sf)

Look again at the original arrangement of this equation. By rearranging it we have swapped the positions of YZ and cos25. This method works whenever the unknown side is at the bottom of the fraction.

Now have a go at this question, using the above method.

Question

Find PQ, giving your answer correct to 3 sf

image: triangle with 40 degrees

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Answer

The answer is 7.15cm.

We have been given angle Q. Side PR is opposite angle Q and PQ is adjacent to it. We have the opposite and the adjacent so we use the formula:

tan = opposite / adjacent

  • tan40 = 6 / PQ
  • By rearranging the formula we get
  • PQ = 6 / tan40
  • Using a calculator
  • PQ = 7.15 (3 sf)

Finding an angle

Question

Find angle C. Give your answer correct to 1 dp.

image: triangle 2 sides given

Answer

We have been given the lengths of AC and BC and asked to find angle C.

  • AC is adjacent to angle C.
  • BC is the hypotenuse.
  • So we use the formula:
  • cos = adjacent / hypotenuse
  • cosC = 5/7
  • To calculate C we need to find the inverse of cos (INV cos or SHIFT cos):

    C = Inv cos(5/7)

    Using a calculator: C = 44.4°

Make sure that you know how your calculator works.

Use your calculator to find Inv cos(5/7), and check that you get the right answer (44.4°).

If you forget how to do it use this as a test:

The inverse cos of 1 is zero. Inv cos 1 = 0 (because cos0 = 1)

Question
image: triangle with 2 sides given

Find angle Z, giving your answer correct to 1 dp.

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Answer

Z = 38.7°

You have been given the opposite and the hypotenuse. Therefore:

sin Z = 5/8

Z = inv sin(5/8)

Z = 38.7° (1 dp)

Questions involving trigonometry and right-angled triangles will only require you to know these three formula (sin, cos and tan). Make sure that you are confident with them.

Activity

Practise recognising the sides of a triangle, and calculating their lengths in this activity.

Now try a Test Bite

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