**Trigonometry: Angles**

**Trigonometry** is a branch of mathematics
dealing with the measurement of sides and **angles** of a triangle. We apply
trigonometry in Engineering, Surveying, Astronomy, Geology etc.

**Angles:**

Let O be the fixed point on OX, the
initial line. Let OY be the revolving line. Then the amount of rotation of OY
about O with respect to OX is known as the **angle** between OX and OY. Here the
angle formed is ∠XOY.

If the revolving line rotates about the fixed
point O in the anticlockwise direction, the angle so formed is said to be
positive.

If the revolving line rotates about the
fixed point O in the clockwise direction, the angle formed is said to be
negative.

**Measurement of Angles**

A line making one complete rotaion makes
360°. When a line makes a quarter tern, it makes 90° or 1 right angle. The size
of a right angle is same in every measurement. The following three system are
commonly used in the measurement of angles:

(a) Sexagesimal System (Degree System)

(b) Centesimal System (Grade System)

(c) Circular System (Radian System)

(a) __Sexagesimal System__: This
system is also called British System. In this system, the unit of measurement
is degree. So, this system also is known as the degree system. In this system, a
right angle is divided into 90 equal parts and each part is called a degree. A
degree is divided into 60 equal parts and each part is called as one minute. A
minute is also divided into 60 equal parts and each part is called as one second.
Therefore, we have

60 seconds = 1 minute (60’’ = 1’)

60 minutes = 1 degree (60’ = 1°)

90 degrees = 1 right angle

The degree, minute and
second are denoted by (°), (’) and (’’) respectively.

(b) __Centesimal System__: This
system is also called the French System. In this system, the unit of measurement
is grade. So, this system also is known as the grade system. In this system, a
right angle is divided into 100 equal parts and each part is called a grade. A
grade is divided into 100 equal parts and each part is called a minute. A
minute is also divided into 100 equal parts and each part is called a second.
Therefore, we have

100 seconds = 1
minute (100’’ = 1’)

100 minutes = 1
grade (100’ = 1^{g})

100 grades = 1
right angles

The grade, minute and
second are denoted by (^{g}), (‘) and (‘’) respectively.

(c) __Circular System__: In this
system, the unit of measurement of an angle is a radian. An angle at the centre
of a circle subtended by an arc equal to the length of radius of the circle is
known as 1 radian. It is denoted by (^{c}).

As the total length of circumference of a
circle is 2πr units, the
angle subtended by circumference of a circle at the centre is 2πr/r radian i.e. 2π^{c}.

Which is, 4 right
angle = 2π^{c}

or, 1 right angle = (π/2)^{c}

Now, from the definition of sexagesimal
measure, centicimal measure and circular measure of angles, we have,

1
right angle = 90° = 100^{g} = (π/2)^{c}

**Theorem: “Radian is a constant angle.”**

Proof:-

Let, O be the centre of the circle and OP = r be the radius of the circle. An arc PQ = r is taken. PO and QO are joined. Produce PO to meet circle at R. Then by definition ∠POQ = 1 radian. The diameter PR = 2r, ∠POR = 2 right angles = 180° and arc PQR = ½ × circumference = ½ × 2πr = πr.

Now, since the angles at the centre of a
circle are proportional to the corresponding arcs on which the stand.

i.e. ∠POQ/∠POR = arc PQ/arc PQR

or, 1
radian/180° = r/πr

or, 1
radian = (180/π)°

Since 1 radian is independent of the
radius of the circle, it is a constant angle.

Proved.

**Relation between different system of measurement of angles:**

Since, 1 right angle = 90° and 1 right
angle = 100^{g}

∴ 90° = 100^{g}

∴ 1° = (10/9)^{g}

Also, 1^{g}
= (9/10)°

Again, π^{c} = 180° = 200^{g}

∴ 1^{c} = (180/π)° and 1^{c} =
(200/π)^{g}

Also, 1°
= (π/180)^{c} and 1^{g}
= (π/200)^{c}

*Workout Examples*

*Workout Examples*

*Example 1: Reduce 24 ^{g} 20’ 44’’ into centicimal seconds.*

*Solution:*

* The given angle
is 24 ^{g} 20’ 44’’*

* = 24 × 100 ×
100’’ + 20 × 100’’ + 44’’*

* = 240000’’ +
2000’’ + 44’’*

* = 242044’’*

*Example 2: Express 42° 20’ 15’’ into the number of degrees.*

*Solution:*

* The given angle
is 42° 20’ 15’’*

* = 42° + (20/60)°
+ (15/60×60)°*

* = 42° + 0.333333°
+ 0.004166°*

* = 42.337499°*

*Example 3: Express 48 ^{g} 54’ 68’’ into degrees, minutes and
seconds.*

*Solution:*

* The given angle
is 48 ^{g} 54’ 68’’*

* = (48 + 54/100 +
68/100×100) ^{g}*

* = (48 + 0.54 +
0.0068) ^{g}*

* = 48.5468 ^{g}*

* = (48.5468 ×
9/10)° [**∵** 1 ^{g} =
(9/10)°]*

* = 43.69212°*

* = 43° + 0.69212°*

* = 43° + (0.69212
× 60)’*

* = 43° + 41.5272’*

* = 43° + 41’ +
0.5272’*

* = 43° + 41’ +
(0.5272 × 60)’’*

* = 43° + 41’ +
31.632’’*

* = 43° 41’ 31.63’’*

*Hence, 48 ^{g} 54’ 68’’ = 43° 41’ 31.63’’*

*Example 4: Express **π**/6 radians into sexagesimal and centicimal measures.*

*Solution:*

* The given angle
is **π**/6 radians*

* = (**π**/6 × 180/**π**)° [**∵** 1 ^{c} = (180/*

*π*

*)°]** = 30°*

*Again,*

* **π**/6 radians*

* = (**π**/6 × 200/**π**) ^{g} [*

*∵*

*1*^{c}= (200/

*π*

*)*^{g}]* = 33.33 ^{g}*

*Example 5: Reduce the following angles into radian measure.*

*a.
**42 ^{g}75’*

*b.
**42° 15’ 30’’*

*Solution:*

*a.
**42 ^{g} 75’*

* = (42 + 75/100) ^{g}*

* = (42 + 0.75) ^{g}*

* = 42.75 ^{g}*

* = (42.75 × **π**/200) ^{c} [*

*∵*

*1*^{g}= (

*π*

*/200)*^{c}]* = 0.21375**π*^{c}

*b.
**42° 15’ 30’’ *

*= (42 +
15/60 + 30/60×60)°*

*= (42 + 0.25
+ 0.00833)°*

*= 42.25833°*

*= (42.25833
× **π**/180) ^{c} [*

*∵*

*1° = (*

*π*

*/180)*^{c}]*= 0.2348**π*^{c}

*Example 6: The angles of a triangle are (7x/2) ^{g}, (9x/4)°
and (*

*π*

*x/50)*^{c}. Find the angles of the triangle in degrees.*Solution:** Here,*

* First angle =
(7x/2) ^{g} = (7x/2 × 9/10)° = (63x/20)°*

* Second angle =
(9x/4)°*

* Third angle = (**π**x/50) ^{c} = (*

*π*

*x/50 × 180/*

*π*

*)° = (18x/5)°**Now, the sum of all angles of a triangle is 2 right angles.*

*i.e. (63x/20)° +
(9x/4)° + (18x/5)° = 180°*

*or, {(63x + 45x +
72x)/20}° = 180°*

*or, (180x/20)° = 180°*

*or, (9x)° = 180°*

*or, x° = 20°*

*Hence, the angles of the triangle are (63×20/20)°, (9×20/4)° and
(18×20/5)° i.e 63°, 45° and 72°.*

*Example 7: Sum of the first and the second angle of a triangle is
150°. The ratio of the number of grades in the first angle to the number of
degrees in the second angle is 5:3. Find the angles of the triangle in circular
measure.*

*Solution:** Suppose number of
grades in the first angle be 5x and the number of degrees in the second angle
be 3x.*

* First angle = 5x ^{g}
= (5x × 9/10)° = (9x/2)°*

* Second angle =
3x°*

*From question,*

* 9x/2 + 3x = 150°*

*or, 15x/2 = 150°*

*or, 15x = 300°*

*or, x = 20°*

*∴** The first angle = 9x/2 = 9×20/2 = 90°*

* The second angle
= 3x = 3 × 20 = 60°*

* The third angle =
180° - 150° = 30°*

*The angles of the triangle in circular measure are,*

* (90 × **π**/180) ^{c}, (60 ×
*

*π*

*/180)*^{c}and (30 ×

*π*

*/180)*^{c}*i.e. (**π**/2) ^{c}, (*

*π*

*/3)*^{c}and (

*π*

*/6)*^{c}*You can comment your
questions or problems regarding the system of measurement of angles here.*

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