Trigonometric functions and circular measure - the Australian ...

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{30} • Trigonometric functions and circular measureStudents should have a deal of practice in finding the trigonometric functions of anglesexpressed in radians. Since students are more familiar with degrees, it is often best toconvert back to degrees.ExampleFind1 cos 4π 32 sin 7π 63 tan 5π 4 .Solution1 cos 4π 3 = cos240◦ = −cos60 ◦ = − 1 22 sin 7π 6 = sin210◦ = −sin30 ◦ = − 1 23 tan 5π 4 = tan225◦ = tan45 ◦ = 1Note. You can enter radians directly into your calculator to evaluate a trigonometric functionat an angle in radians, but you must make sure your calculator is in radian mode.Students should be reminded to check what mode their calculator is in when they aredoing problems involving the trigonometric functions.Arc lengths, sectors and segmentsMeasuring angles in radians enables us to write down quite simple formulas for the arclength of part of a circle and the area of a sector of a circle.In any circle of radius r , the ratio of the arc length l to the circumference equals the ratioof the angle θ subtended by the arc at the centre and the angle in one revolution.lθrO
A guide for teachers – Years 11 and 12 • {31}Thus, measuring the angles in radians,l2πr = θ2π=⇒ l = r θ.It should be stressed again that, to use this formula, we require the angle to be in radians.ExampleIn a circle of radius 12 cm, find the length of an arc subtending an angle of 60 ◦ at thecentre.O60 º12 cmSolutionWith r = 12 and θ = 60 ◦ = π , we have3l = 12 × π = 4π ≈ 12.57 cm.3It is often best to leave your answer in terms of π unless otherwise stated.We use the same ratio idea to obtain the area of a sector in a circle of radius r containingan angle θ at the centre. The ratio of the area A of the sector to the total area of the circleequals the ratio of the angle in the sector to one revolution.Thus, with angles measured in radians,Aπr 2 = θ2π=⇒ A = 1 2 r 2 θ.The arc length and sector area formulas given above should be committed to memory.
Page 1 and 2: 1Supporting Australian Mathematics
Page 3 and 4: Assumed knowledge . . . . . . . . .
Page 5 and 6: A guide for teachers - Years 11 and
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