Trigonometry: Radian Measure

TRIGONOMETRY – EDEXCEL SYLLABUS SAYS:

"Radian measure, including use of arc length and area of sector. The formulae s = rӨ and A = ½ r²Ө for a circle are expected to be known."

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Radian Measure

1)      What is a radian? How to convert degrees to radian and vice versa?

a.       1 radian subtends an arc length of 1 radius, r, in a sector; or, conversely,
       an arc length of 1 radius in a sector subtends an angle of 1 radian at the centre of the circle.

b.      Since a circle has 2πr as its ‘arc length’ or, circumference,

                                                               i.      a circle of 360^o has (2πr ÷ r) radian or 2π radian;

                                                             ii.      which means 360^o = 2π radian

                                                            iii.      180^o = π radian (2π/2 radian)

                                                           iv.      90^o = π/2 radian (since 2 x 90^o = 180^o)

                                                             v.      60^o = π/3 radian (since 3 x 60^o = 180^o)

                                                           vi.      45^o = π/4 radian (since 4 x 45^o = 180^o)

                                                          vii.      30^o = π/6 radian (since 6 x 30^o = 180^o)

                                                        viii.      270^o = 3π/2 radian (since 3 x 90^o = 180^o)

c.       Generally, to convert x^o to radian, we use:

   

x^o = (x^o/180^o) π radian

d.       And, to convert Ө radian to degrees, we use:

   

Ө radian = (Ө/π) 180^o (since π radian = 180^o)

2)      Arc Length, s = Өr

a.       Arc Length: Since 1 radian subtends an arc length of 1 radius, r, Ө radian therefore subtends Өr arc length. Thus,

Arc Length, s = Өr

b.      Segment perimeter, p = Өr + 2r sin(Ө/2)

                                                              i.      Segment perimeter = arc length + chord length

                                                            ii.      Arc length = Өr

                                                          iii.      Chord length = (2)[r sin(Ө/2)] --- bisect Ө to find half-chord length, then multiply by 2 to get full-chord length

                                                          iv.      Thus, Segment perimeter, p = Өr + 2r sin(Ө/2)

3)      Area of sector, A_sec = (1/2)(Ө)r²

Sector Area

a.       Since Circle Area = πr²;

b.      A Circle has 360^o or 2π radian at its centre.

c.       So, its sector of Ө radian will have a sector area A = (Ө/2π) πr² proportionately. After cancelling down or simplifying, we get:

Sector area, A_sec = (1/2)(Ө)r²

 

                  Segment Area, A_seg = (1/2)r²(Ө - sinӨ)

d.      Segment area = Sector Area – Triangle Area (the triangle formed by the 2 radii and the chord of the sector)

e.       Sector area, A_sec = (1/2)(Ө)r² – please see above

f.        Triangle Area, A_Δ = (1/2)(r)(r)sinӨ – Sine Formula for Area of Δ

g.       Therefore,

Segment Area = Sector Area – Triangle Area

                        = (1/2)(Ө)r² - (1/2)(r)(r)sinӨ

                        = (1/2)r²(Ө – sinӨ)

(Practice Questions coming up soon...Next, the relationship between Reference Angle, Principal Angle and the angle that you want to find...Click and view this video as it will help you in this area.)

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