Trigonometric functions and circular measure - the Australian ...

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{28} • Trigonometric functions and circular measureHence, if we are interested only in the acute angle, since tan(180 ◦ − θ) = −tanθ, we cantake the absolute value and say that, if γ is the acute angle between the two lines, then∣ tanγ =m 1 − m 2 ∣∣∣∣,1 + m 1 m 2provided the lines are not perpendicular.ExampleFind, to the nearest degree, the acute angle between the lines y = 2x − 1 and y = 3x + 4.SolutionHere m 1 = 2 and m 2 = 3. So, if θ is the acute angle between the two lines, we havetanθ =2 − 3∣1 + 6∣ = 1 7and therefore θ ≈ 8 ◦ .Exercise 14Find the two values of m such that the angle between the lines y = mx and y = 2x is 45 ◦ .What is the relationship between the two lines you obtain?If we define the angle between two curves at a point of intersection to be the angle betweentheir tangents at that point, then the above formula — along with some differentialcalculus — can be used to find that angle.Radian measureThe measurement of angles in degrees goes back to antiquity. It may have arisen from theidea that there were roughly 360 days in a year, or it may have arisen from the Babylonianpenchant for base 60 numerals. In any event, both the Greeks and the Indians dividedthe angle in a circle into 360 equal parts, which we now call degrees. They further dividedeach degree into 60 equal parts called minutes and divided each minute into 60 seconds.An example would be 15 ◦ 22 ′ 16 ′′ . This way of measuring angles is very inconvenient andit was realised in the 16th century (or even before) that it is better to measure angles viaarc length.We define one radian, written as 1 c (where the c refers to circular measure), to be theangle subtended at the centre of a unit circle by a unit arc length on the circumference.
A guide for teachers – Years 11 and 12 • {29}y(0,1)1(–1,0)O1(1,0)x(0,–1)Definition of one radian.Since the full circumference of a unit circle is 2π units, we have the conversion formula360 ◦ = 2π radiansor, equivalently,180 ◦ = π radians.So one radian is equal to 180π degrees, which is approximately 57.3◦ .Since many angles in degrees can be expressed as simple fractions of 180, we use π as abasic unit in radians and often express angles as fractions of π. The commonly occurringangles 30 ◦ , 45 ◦ and 60 ◦ are expressed in radians respectively as π 6 , π 4 and π 3 .ExampleExpress in radians:1 135 ◦ 2 270 ◦ 3 100 ◦ .Solution1 135 ◦ = 3 × 45 ◦ = 3π 4c2 270 ◦ = 3 × 90 ◦ = 3π 2c3 100 ◦ = 100π c= 5π c180 9Note. We will often leave off the c, particularly when the angle is expressed in terms of π.
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