Mathematics Angles, Degree measure ,Radian measure and Relation between degree and radian
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Topics Covered

star Introduction
star Angles
star Degree measure
star Radian measure
star Relation between degree and radian

Introduction

\color{green} ✍️ The word "‘trigonometry’" is derived from the Greek words color{green}{"‘trigon’"} and color{green}{"‘metron’"} and it means color{green}{"‘measuring the sides of a triangle’."} The subject was originally developed to solve geometric problems involving triangles.

\color{green} ✍️ we will generalise the concept of trigonometric ratios to trigonometric functions and study their properties.

Angles

\color{green} ✍️ Angle is a measure of rotation of a given ray about its initial point.

\color{green} ✍️ The original ray is called color{blue}{ul"the initial side"} and the final position of the ray after rotation is called color{blue}{ul"the terminal side of the angle .")

\color{green} ✍️ The point of rotation is called color{blue}{ul"the vertex."}

\color{green} ✍️ If the direction of rotation is anticlockwise, the angle is said to be color{blue}{ul"positive") and if the direction of rotation is clockwise, then the angle is color{blue}{ul"negative"} (Fig 3.1).

Degree measure

● The measure of an angle is the amount of rotation performed to get the terminal side from the initial side.

● There are several units for measuring angles.

●The definition of an angle suggests a unit, viz. one complete revolution from the position of the initial side as indicated in Fig opposite

● If a rotation from the initial side to terminal side is (1/360)^(th) of a revolution, the angle is said to have a measure of color{blue}{ul"one degree"}, written as 1°.

● A degree is divided into 60 minutes, and a minute is divided into 60 seconds.

● One sixtieth of a degree is called color{blue}{ul"a minute"}, written as 1′.

● one sixtieth of a minute is called color{blue}{ul"a second"}, written as 1″.

Thus,  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color{green} (1° = 60′, \ \ \ \ \ \ \1′ = 60″)

Some of the angles whose measures are 360°,180°, 270°, 420°, – 30°, – 420° are shown in Fig 3.3.

Radian measure

● There is another unit for measurement of an angle, called the color{blue}{ul"radian measure.")

color(blue)(☛ ulmathtt"1 radian")

Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of color{green}(ul"1 radian").

● The figures show the angles whose measures are 1 radian, –1 radian, 1 1/2 radian and –1 1/2 radian.

● We know that the circumference of a circle of radius 1 unit is 2π. Thus, one complete revolution of the initial side subtends an angle of 2π radian.

● A circle of radius r, an arc of length r subtends an angle whose measure is 1 radian, an arc of length l will subtend an angle whose measure is l/r radian.

\color{green} ✍️ Thus, if in a circle of radius r, an arc of length l subtends an angle θ radian at the centre,

we have color{green}(θ = l/r) or color{green}(l = r θ.)
Q 3126301271

Find the radius of the circle in which a central angle of 60° intercepts an arc of length 37.4 cm (use pi = 22/7)

Solution:

Here l = 37.4 cm and θ = 60° = (60pi)/(180) radain = pi/3

Hence, by r = l/theta, we have

r = (37.4xx3)/(pi) = (37.4xx3xx7)/(22) = 35.7 cm
Q 3156301274

The minute hand of a watch is 1.5 cm long. How far does its tip move in 40 minutes? (Use π = 3.14).

Solution:

In 60 minutes, the minute hand of a watch completes one revolution. Therefore in 40 minutes, the minute hand turns through 2/3 of a revolution. Therefore theta = 2/3 xx360° or (4pi)/3 Hence, the required distance travelled is given by

l = r θ = 1.5 xx (4pi)/3 cm = 2pi cm = 2 × 3.14 cm = 6.28 cm.
Q 3176301276

If the arcs of the same lengths in two circles subtend angles 65°and 110° at the centre, find the ratio of their radii.

Solution:

Let r_1 and r_2 be the radii of the two circles. Given that

θ_1 = 65° = pi/180 xx 65 = (13pi)/36 radian

and θ_2= 110° pi /1800 xx 100 = (22pi)/36 radian

Let l be the length of each of the are. Then l = r_1θ_1 = r_2θ_2, which gives

(13pi)/36 xxr_1 = (22pi)/36 xxr_2 , i.e (r_1)/(r_2) = 22/13

Hence r_1 : r_2 = 22:13.

Relation between degree and radian

● Since a circle subtends at the centre an angle whose radian measure is 2π and its degree measure is 360°, it follows that

 \ \ \ \ \ \ \ \ \ \ \color{blue}("2π radian" = 360°) or  \ \ π radian = 180°

● Using approximate value of π as 22/ 7 , we have

 \ \ \ \ \ \ \ \ 1 radian = (180°)/ π = 57° 16′ approximately.

Also 1° = π /180 radian = 0.01746 radian approximately.

The relation between degree measures and radian measure of some common angles are given in the following table:

\color{fuchsia} { ul ☛ "Notational Convention"}

 \ \ \ color{blue} {π= 180°} and π/4= 45° are written with the understanding that π and π/4 are radian measures.

Thus, we can say that

color{blue}{"Radian measure" = π/ 180 × "Degree measure"}

color{blue}{"Degree measure "= 180 /π ×" Radian measure"}
Q 3166201175

Convert 40° 20′ into radian measure.

Solution:

We know that 180° = π radian.
Hence 40° 20′ = 40 1/3 degree = pi/180 xx 121/3 radian = (121pi)/(540) radian

Therefore 40° 20′ (121pi)/(540) radian