GCSE Trigonometry of Right-Angled Triangles

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 1^st January 2019

Objectives:

• Find unknown sides in right-angled triangles using a side and angle.
• Find unknown angles in right-angled triangle using two sides.
• Note: ‘Exact trigonometric ratios’ and 3D Trigonometry are covered in separate slides.

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Frost Childhood Story

I was in Year 9 and was trying to write a computer program that would draw an analogue clock (as you do).

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I needed to work out the two coordinates to draw the minute hand between, and similarly for the hour hand, given the current hour, minute and desired length of each hand.

I couldn’t work out how to do it, until I (coincidentally) learnt trigonometry in maths the next day, and was able to finish my program that evening!

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We used Pythagoras’ theorem if all 3 sides of a right-angled triangle were involved.

But what if we had two sides involved and an angle? (excluding the right-angle)

Unknown angle.

Unknown side.

Unknown side.

30°

hypotenuse

adjacent

opposite

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The ‘adjacent’ is the side adjacent (i.e. next to) the angle of interest.

The ‘opposite’ is the side opposite the angle of interest.

The hypotenuse is the longest side of a right-angled triangle, and is opposite the right-angle.

60°

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20°

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🖉 sin, cos and tan are functions which take an angle and give us the ratio between pairs of sides in a right angle triangle.

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Recall that ratio can just mean how many times bigger one quantity is than another.

Step 1: Determine which sides are hyp/adj/opp.

Step 2: Work out which trigonometric function we need.

Step 3: Rearrange the equation to find the unknown.

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(questions on provided sheet)

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Q2-7 on next slide…

(questions on provided sheet)

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[Edexcel GCSE June2012-2H Q18] 

The diagram shows a quadrilateral ABCD. AB = 16 cm. AD = 12 cm. Angle BCD = 40°.�Angle ADB = angle CBD = 90°.

Calculate the length of CD.

16.5 cm

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[OCR GCSE(9-1) SAM 6H Q16aii]

Simon cuts the corners off a square piece of card to leave the regular octagon shown below. �O is the centre of the octagon. A and B are vertices of the octagon. OA = OB = 5 cm. Angle AOB = 45°.

Work out the area of the original square piece of card. 85.4 cm²

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Frost Childhood Story

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When would we use trigonometry?

• When we have a right-angled triangle.
• When we’re involving two side lengths and an angle.

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Step 1: Determine which sides are hyp/adj/opp.

Step 2: Work out which trigonometric function we need.

Step 3: Rearrange the equation to find the unknown.

We can use the same process…

Remember that the angle always goes after the sin/cos/tan.

What is the missing angle?

What is the missing angle?

What is the missing angle?

What is the missing angle?

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[Edexcel IGCSE Jan2014(R)-3H Q17] The diagram shows triangle ABC. D is the point on AB, such that CD is perpendicular to AB.�AC = 8.3 cm. AD = 4.7 cm. BD = 7.5 cm.

Calculate the size of angle ABC.�Give your answer correct to 1 decimal place.

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[Edexcel GCSE June2014-2H Q15b] The diagram shows the positions of three turbines A, B and C.

Calculate the bearing of C from A.

Give your answer correct to the nearest degree.

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