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© Boardworks Ltd 2005 1 of 40 © Boardworks Ltd 2005 1 of 40 AS-Level Maths: Core 2 for Edexcel C2.5 Trigonometry 2 This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation.

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© Boardworks Ltd 2005 2 of 40 Contents © Boardworks Ltd 2005 2 of 40 The sine rule The cosine rule The area of a triangle using ½ ab sin C Degrees and radians Arc length and sector area Solving equations using radians Examination-style questions Arc length and sector area

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© Boardworks Ltd 2005 3 of 40 Using radians to measure arc length Suppose an arc AB of a circle of radius r subtends an angle of θ radians at the centre. r r θ O A B where θ is measured in radians. If the angle at the centre is 1 radian then the length of the arc is r. If the angle at the centre is 2 radians then the length of the arc is 2 r. If the angle at the centre is 0.3 radians then the length of the arc is 0.3 r. In general: Length of arc AB = θr When θ is measured in degrees the length of AB is

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© Boardworks Ltd 2005 4 of 40 Finding the area of a sector We can also find the area of a sector using radians. where θ is measured in radians. r r θ O A B Again suppose an arc AB subtends an angle of θ radians at the centre O of a circle. The angle at the centre of a full circle is 2 π radians. When θ is measured in degrees the area of AOB is In general: Area of sector AOB = r 2 θ Area of sector AOB = So the area of the sector AOB is of the area of the full circle.

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© Boardworks Ltd 2005 5 of 40 Finding chord length and sector area A chord AB subtends an angle of radians at the centre O of a circle of radius 9 cm. Find in terms of π : a) the length of the arc AB. b) the area of the sector AOB. 9 cm O A B a) length of arc AB = θr = 6 π cm = 27 π cm 2 b) area of sector AOB = r 2 θ

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© Boardworks Ltd 2005 6 of 40 Finding the area of a segment A chord AB divides a circle of radius 5 cm into two segments. If AB subtends an angle of 45° at the centre of the circle, find the area of the minor segment to 3 significant figures. 5 cm 45° O A B The formula for the area of a sector can be combined with the formula for the area of a triangle to find the area of a segment. For example: Let’s call the area of sector AOB A S and the area of triangle AOB A T.

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© Boardworks Ltd 2005 7 of 40 Finding the area of a segment Now: Area of the minor segment = A S – A T = 9.8174… – 8.8388… = 0.979 cm 2 (to 3 sig. figs.) In general, the area of a segment of a circle of radius r is: where θ is measured in radians.

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O A B AOB = Sector / Juring = Arc / Busur ARC SECTOR Arc lengths and Areas of Sectors Created by ﺠﻴﻄ for Mathlabsky.wordpress.com.

O A B AOB = Sector / Juring = Arc / Busur ARC SECTOR Arc lengths and Areas of Sectors Created by ﺠﻴﻄ for Mathlabsky.wordpress.com.

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