Radian Measure
Trigonometry: Radians KUS objectives BAT convert fluently between degrees and radians BAT calculate arcs, sectors and segments Starter: Simplify (without a calculator) sin 30 1+ 3 + 3 cos 60 2 sin 45 + cos 45 1+ 2 tan 150 1+ 3 − 3 tan 120
Introduction to Radian measure1 For a circle of radius r, the angle at the centre, which subtends an arc of length r, is equal to 1 radian ‘If arc AB has length r, then angle AOB is 1 radian (1c or 1 rad)’ As the circumference of the circle is 2πr the total angle is 2π radians Radians are an alternative to degrees. Some calculations involving circles are easier when Radians are used, as opposed to degrees. r A B O 1c 45 135 225 0 30 60 90 120 150 180 210 𝜋 𝑐 0 𝑐
WB 1 Convert the following angles to degrees Converting radians and degrees Radians Degrees 𝜋 𝑐 =180 WB 1 Convert the following angles to degrees 7𝜋 8 𝑐 4𝜋 15 𝑐 a) 1.8 𝑐 b) c) =1.8 × 180 𝜋 = 7𝜋 8 × 180 𝜋 = 4𝜋 15 × 180 𝜋 =103° = 1260𝜋 8𝜋 =157.5° = 720𝜋 15𝜋 =48°
Now do skills 249 35 0 =110× 𝜋 180 =150× 𝜋 180 =35× 𝜋 180 = 5𝜋 6 𝑐 Converting degrees to radians Degrees Radians 𝜋 𝑐 =180 WB 1 Convert the following angles to radians d) e) f) 35 0 =150× 𝜋 180 =110× 𝜋 180 =35× 𝜋 180 = 5𝜋 6 𝑐 = 11𝜋 18 = 7𝜋 36 = 0.611 𝑐 The superscript 𝑐 is often unwritten Now do skills 249
ARC Length
This is the formula’s usual form SECTOR area The Area of a Sector and Segment can be worked out using Radians = A = Multiply by π O θ X = Multiply by r2 B = = This is the formula’s usual form
r= 10 𝜃 + 2 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟=𝑟𝜃+2𝑟 10=𝑟(𝜃+2) 10 𝜃+2 =𝑟 WB 2a arc length Arc AB of a circle, with centre O and radius r, subtends an angle of θ radians at O. The Perimeter of sector AOB is 10 cm. Express r in terms of θ. Length AB = rθ 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟=𝑟𝜃+2𝑟 10=𝑟(𝜃+2) 10 𝜃+2 =𝑟 r θ A B O rθ r= 10 𝜃 + 2
Finding the length of an arc is easier when you use radians WB 3 arc length Calculator in Radians Finding the length of an arc is easier when you use radians The border of a garden pond consists of a straight edge AB of length 2.4m, and a curved part C, as shown in the diagram below. The curved part is an arc of a circle, centre O and radius 2m. Find the length of C. O (H) 2m x A 1.2m B Inverse sine (O) Double for angle AOB O A B 2.4m 2m C θ Angle θ = 2π – 1.287 Angle θ = 4.996 rad (We need to work out angle θ)
WB 4 sector area In the diagram, the area of the minor sector AOB is 28.9cm2. Given that angle AOB is 0.8 rad, calculate the value of r. Put the numbers in ½ x 0.8 = 0.4 B O 0.8c A r cm Divide by 0.4 Square root
WB 5 sector area A plot of land is in the shape of a sector of a circle of radius 55m. The length of fencing that is needed to enclose the land is 176m. Calculate the area of the plot of land. The length of the arc must be 66m (adds up to 176 total) Put the numbers in Divide by 55 A 55m 66m 1.2c θ O Put the numbers in 55m B (We need to work out the angle first) Now do skills 251
It is not necessary too memorise this formula – use common sense Segment area Work out the area of a segment using radians. Area of a Segment Area of Sector AOB – Area of Triangle AOB Area of Sector AOB Area of Triangle AOB O r r θ a = b = r C = θ A B Area of the Segment Factorise It is not necessary too memorise this formula – use common sense
Calculate the Area of the segment shown in the diagram below. WB 6 segment area Calculate the Area of the segment shown in the diagram below. Substitute the numbers in Work the parts out O π 3 2.5cm Only round the final answer
Remember, sin x = sin (180 – x) WB 7 segment area In the diagram AB is the diameter of a circle of radius r cm, and angle BOC = θ radians. Given that the Area of triangle AOC is 3 times that of the shaded segment, show that 3θ – 4sinθ = 0 Area of the shaded segment Area of triangle AOC a = b = r Angle = π-θ Remember, sin x = sin (180 – x) A C B θ AOC = 3 x shaded segment Cancel out 1/2r2 Multiply out the brackets Subtract sinθ
WB 8 Figure 2 shows a plan of a patio. The patio PQRS is in the shape of a sector of a circle with centre Q and radius 6 m. Given that the length of the straight line PR is 63 m, (a) find the exact size of angle PQR in radians. (b) Show that the area of the patio PQRS is 12 m2. (c) Find the exact area of the triangle PQR. (d) Find, in m2 to 1 decimal place, the area of the segment PRS. (e) Find, in m to 1 decimal place, the perimeter of the patio PQRS.
WB 9
WB 9 exam Q - solution
WB 10
One thing to improve is – KUS objectives BAT convert fluently between degrees and radians BAT calculate arcs, sectors and segments self-assess One thing learned is – One thing to improve is –
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