# Conversion from Sexagesimal to Circular System

Worked-out problems on the conversion from sexagesimal to circular system:

1. Express 40° 16’ 24” is radian.

Solution:

40° 16’ 24”

= 40° + 16’ + 24”

We know 1° = 60”

= 40° + 16’ + (24/60)’

= 40° + (16 + 2/5)’

= 40° + (82/5)’

We know 1° = 60’

= 40° + (82/5 × 60)°

= (40 + 41/150)°

= (6041/150)°

We know 180° = πc

Therefore, 6041°/150 = (πc/180) × (6041/150) = 6041/27000 πc

Therefore, 40° 16’ 24” = 6041/27000 πc

2. Show that 1° < 1c

Solution:

We know 180° = πc

or, 1° = (π/180)c

or, 1° = (22/7 × 180) c < 1c

Therefore, 1° < 1c

3. Two angles of a triangle are 75° and 45°. Find the value of the third angle in circular measure.

In ∆ABC, ∠ABC = 75° and ∠ACB = 45°; ∠BAC = ?

You know that the sum of the three angles of a triangle is 180°

Therefore, ∠BAC = 180° - (75° + 45°)

= 180° - 120°

= 60°

Again, we know: 180° = π

Therefore, 60° = 60 π/180 = π/3

In ΔABC, ∠BAC = π/3

4. A rotating ray revolves in the anticlockwise direction and makes two complete revolutions from its initial position and moves further to trace an angle of 30°. What are the sexagesimal and circular measures of the angle with reference to trigonometrical measure?

As the rotating ray does in the anti-clockwise direction, the angle formed is positive. We know, in one complete revolution the rotating ray traces an angle of 360°. So in two complete revolutions it makes an angle of 360° × 2 i.e. 720°. It has moved further to trace an angle of 30°. So the magnitude of the angle formed is (720° + 30°) i.e. 750°

Now, 180° = π

Therefore, 750° = 750 π/180 = 25 π/6

5. The ratio of the angles subtended at the centre by two unequal arcs of a circle is 5 : 3. If the magnitude of the second angle is 45°, find the sexagesimal and circular measures of the first angle.

Let the measure of the first angle be θ°

Then, according to the given condition, θ°/45° = 5/3

Therefore, θ° = 5/3 × 45° = 75°

Again we know, 180° = π

Therefore, 75° = 75 π/180 = 5 π/12

Therefore, the sexagesimal measure of the first angle is 75° and circular measure is 5 π/12.

6. ABC is an equilateral triangle in which AD is the line segment that joins the vertex A to the mid point of the side BC. What is the circular measure of ∠BAD?

Solution:

As ∆ABC is equilateral

Therefore, ∠BAC = 60°

We also know that the median of an equilateral triangle bisects the corresponding vertiealange. Therefore, ∠BAD = 30°

Therefore, the circular measure of ∠BAD = 30 π/180 = π/6

The above solved problems help us to learn in trigonometry, about the conversion from sexagesimal to circular system.

Basic Trigonometry

Trigonometry

Circular System

Relation between Sexagesimal and Circular

Conversion from Sexagesimal to Circular System

Conversion from Circular to Sexagesimal System