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Published byCarter Corcoran Modified over 2 years ago

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RADIANS Definition An arc of length r subtends an angle of one radian at the centre of a circle of radius r r r r Ø=1radian

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Proof r r r How do you calculate the length of an arc? r = ø x 2πr 360º r = 1 radian x 2πr 360 º 360r = 1 radian x 2πr x 360 º 360 º = 1 radian x 2π ÷ r ÷ 2180 º = 1 radian x π ÷ π 180 = 1 radian π Or 180 º = π radians so 1 radian is approximately…?

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Converting between angles and radians Degrees = radians x 180 π Radians = degrees x π So if is measured in radians So if ø is measured in radians Then radians = x π Then ø radians = ø x π

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How many different angles can you write as radians? 180º = π radians 90º = π radians 2

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Arc Length r r r Ø=1radian Arc length = ø x 2πr 360º Arc length = 2πrø 360º Arc length = r 2πø 360º Factorise r Divide by 2 Arc length = r πø 180º Angle in radians Arc length = rø Angle in degrees

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Area of Sector r r r Ø=1radian Sector area = ø x πr 2 360º Sector area = πr 2 ø 360º Sector area = r 2 πø 360º Factorise r 2 Factorise out ½ Sector area = ½r 2 πø 180º Angle in radians Sector area = ½r 2 ø Angle in degrees

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Examples Convert 50° into radians 50° = 50° x π rad 50° = 50° x π rad ° = 0.87 rad 50° = 0.87 rad

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Examples Convert 2.7 radians into degrees 2.7 rad = 2.7 x 180 degrees 2.7 rad = 2.7 x 180 degrees π 2.7 rad = °

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Examples Convert 40° into radians 40° = 40 x π 40° = 40 x π ° = 40π 40° = 40π ° = 2π radians 40° = 2π radians 9

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Calculate the arc length and sector area 10cm 1.2radians Arc length = rө Arc length = 10 x 1.2 Arc length = 12cm Sector area = ½r 2 ө Area = ½ x 100 x 1.2 Area = 60cm 2

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