GCSE Trigonometry of Right Angled Triangles Dr J

GCSE Trigonometry of Right. Angled Triangles Dr J Frost (jfrost@tiffin. kingston. sch. uk) 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 3 D Trigonometry are covered in separate slides. Last modified: 1 st January 2019

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

Starter 13 3 5 4 ? ?

But… We used Pythagoras’ theorem if all 3 sides of a right-angled triangle were involved. But what if we had two sides involved an angle? (excluding the right-angle) Unknown angle. Unknown side.

Names of sides relative to an angle The hypotenuse is the longest side of a rightangled triangle, and is opposite the right-angle. opposite ? The ‘opposite’ is the side opposite the angle of interest. ? hypotenuse 30° adjacent ? The ‘adjacent’ is the side adjacent (i. e. next to) the angle of interest.

Quickfire Side Naming Hypotenuse Opposite Adjacent 60° 1 2 ? ? 1? ? 2 ? ? ? 20°

Sin/Cos/Tan ! sin, cos and tan are functions which take an angle and give us the ratio between pairs of sides in a right angle triangle. Recall that ratio can just mean how many times bigger one quantity is than another. ? ? You can remember this using: “soh cah toa” ?

Examples 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.

Further Example ?

Test Your Understanding So Far ?

Harder Examples 60 ° ? 12 ? 30° 4

Test Your Understanding 1 2 ? 8. 63 ? 20. 2

Exercise 1 1 (questions on provided sheet) a ? b 15 22 c 4 e ? ? f ? 10 20 g ? d ? 3 Q 2 -7 on next slide…

Exercise 1 2 (questions on provided sheet) 5 ? ? 3 6 ? 4 [Edexcel GCSE June 2012 -2 H Q 18] 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 ? [OCR GCSE(9 -1) SAM 6 H Q 16 aii] 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 cm 2 ? N ?

y x

STARTER RECAP When would we use trigonometry? • When we have a right-angled triangle. • When we’re involving two? side lengths and an angle. 4 30 ° ?

But what if the angle is unknown? We can use the same process… 3 Step 1: Determine which sides are hyp/adj/opp. 5 Step 2: Work out which trigonometric function we need. Remember that the angle always goes after the sin/cos/tan. Step 3: Rearrange the equation to find the unknown.

What is the missing angle?

What is the missing angle?

What is the missing angle?

What is the missing angle?

Test Your Understanding [Edexcel IGCSE Jan 2014(R)-3 H Q 17] 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. ? ?

Exercise 2 (questions on provided sheet) 1 a ? c ? 2 [Edexcel GCSE June 2014 -2 H Q 15 b] 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. ? e ? b d ? 3 ?

Exercise 2 (questions on provided sheet) 5 4 ? 6 ? N ? ?

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