Chapter 2: Pythagoras Theorem and Trigonometry (Revision)

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1 Chapter 2: Pythagoras Theorem and Trigonometry (Revision) Paper 1 & 2B 2A Triangles Understand a proof of Pythagoras Theorem. Understand the converse of Pythagoras Theorem. Use Pythagoras Triangles Use Pythagoras Theorem in 3-D situations (e.g. to determine lengths inside a cuboid). Trigonometry Trigonometric ratios Understand, recall and use the trigonometric relationships in right-angled triangles, namely, sine, cosine and tangent. Use the trigonometric ratios to solve problems in simple practical situations (e.g. in problems involving angles of elevation and depression). Trigonometry Trigonometric ratios Extend the use of the sine and cosine functions to angles between 90 and 180. Solve simple trigonometric problems in 3- D. (e.g. find the angle between a line and a plane and the angle between two planes) Sine and cosine rules Use the sine and cosine rules to solve any triangle. 1 P age

2 Section 2.1 Pythagoras Theorem If the triangle had a right angle (90 ) and you made a square on each of the three sides, then the biggest square had the exact same area as the other two squares put together! It is called "Pythagoras' Theorem" and can be written in one short equation: a 2 + b 2 = c 2 Note: c is the longest side of the triangle a and b are the other two sides The longest side of a triangle is always called the HYPOTENUSE. REMEMBER: Pythagoras Theorem can only be used in a right- angled triangle. 2 P age

3 Example 1 Example 2 3 P age

4 Example 3 Example 4 Support Exercise Edexcel Pg 297 Ex 19A No 1 5 Pg 300 Ex 19B No 1 10 Handout Pythagoras Theorem 4 P age

5 Section 2.2 Trigonometric Ratios Trigonometry uses three important ratios to calculate sides and angles: sine, cosine and tangent. These ratios are defined in terms of the sides of a right- angled triangle and an angle. The angle is often written as θ. In a right- angled triangle: the side opposite the right angle is called the hypotenuse and is the longest side the side opposite the angle θ is called the opposite side the other side next to both the right angle and the angle θ is called the adjacent side. The sine, cosine and tangent ratios for θ are defined as: In order to remember these we use the word SOH CAH TOA Sin = Opp / Hyp Cos = Adj / Hyp Tan = Opp / Adj 5 P age

6 Example 1 Example 2 Example 3 6 P age

7 Example 4 A ladder 5 m long, leaning against a vertical wall makes an angle of 65 with the ground. a) How high on the wall does the ladder reach? b) How far is the foot of the ladder from the wall? c) What angle does the ladder make with the wall? Solution: a) The height that the ladder reaches is PQ PQ = sin 65 5 = 4.53 m b) The distance of the foot of the ladder from the wall is RQ. RQ = cos 65 5 = 2.11 m c) The angle that the ladder makes with the wall is angle P 7 P age

8 Support Exercise Edexcel Pg 307 Ex 19D Nos 1 5 Handout 8 P age

9 Section 2.3 Working with Trigonometric Ratios to find Angles Support Exercise Pg 309 Ex 19E Nos 1 4 Handout 9 P age

10 Section 2.4 Mixed Examples Example 1 ABCD is a quadrilateral. Angle BDA = 90, angle BCD = 90, angle BAD = 40. BC = 6 cm, BD = 8cm. a) Calculate the length of DC. Give your answer correct to 3 significant figures. [Hint: use Pythagoras theorem!] b) Calculate the size of angle DBC. Give your answer correct to 3 significant figures. c) Calculate the length of AB. Give your answer correct to 3 significant figures. 10 P age

11 Example 2 AB = 19.5 cm, AC = 19.5 cm and BC = 16.4 cm. Angle ADB = 90 degrees. BDC is a straight line. Calculate the size of angle ABC. Give your answer correct to 1 decimal place. 11 P age

12 Example 3 P l 13 cm S l Q 5 cm R Find the value of l giving your answer to three significant figures. Support Exercise Handout 12 P age