`star` Introduction

`star` Angles

`star` Degree measure

`star` Radian measure

`star` Relation between degree and radian

`star` Angles

`star` Degree measure

`star` Radian measure

`star` Relation between degree and radian

`\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.

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

`\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).

`\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).

● 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.

● 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.

● 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.)`

`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`)

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`).

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.

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.`

● 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"}`

` \ \ \ \ \ \ \ \ \ \ \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.

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