Both sexagesimal and circular are the systems of measuring angles. The unit of angle measurement in the sexagesimal system is degrees, minutes and seconds. In a circular system, the unit of angle measurement in radians. So, here we will learn how to convert degrees, minutes, seconds to radians easily. Solve the example problems to know more about the concept.
Sexagesimal to Circular System Conversion
Sexagesimal systems, circular systems are the different systems of measuring trigonometric angles. To convert from sexagesimal to the circular system, we need to know the relationship between them. The relation between sexagesimal and circular system is 180° = π radians. Get the simple steps to convert the unit of angle from the following sections.
- Convert the minutes and seconds into the degrees.
- Now multiply the degrees by (π/180°).
- The result will be the angle in radians.
Also, Read
Questions on Conversion from Sexagesimal to Circular System
Question 1:
Convert 52° 18′ 24″ to radians.
Solution:
52° 18′ 24″
Convert minutes, seconds into degrees
we know 1° = 60″, 1° = 60′
52° 18′ 24″ = 52° + 18′ + (24/60)’
= 52° + (276/15)’
= 52° + (276/15 x 60)°
= 52° + (23/75)°
= (3923/75)°
We know that 180° = πc
Therefore, (3923/75)° = (3923/75) x (πc/180°)
= 3923/13500 πc
So, 52° 18′ 24″ = 3923/13500 πc
Question 2:
The ratio of the angles subtended at the center by two unequal arcs of a circle is 7:5. If the magnitude of the second angle is 75°, find the sexagesimal and circular measures of the first angle.
Solution:
Let the measure of the first angle be θ°
As per the given condition, θ°/75° = 7/5
θ° = 7/5 x 75°
= 105°
We know that 180° = π
So, 105° = 105 x π/180° = 7π/12
Therefore, the sexagesimal measure of the first angle is 105° and circular measure is 7π/12
Question 3:
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 45°. What are the sexagesimal and circular measures of the angle with reference to trigonometrical measure?
Solution:
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 45°. So the magnitude of the angle formed is (720° + 45°) i.e. 765°
We know that 180° = π
Therefore, 765° = 765 x π/180 = 17π/4
Question 4:
If the two angles of a triangle are 60°, 50°. Find the value of the third angle in circular measure.
Solution:
The sum of the internal angles of a triangle is 180°
So, 60° + 50°+ third angle = 180°
third angle = 70°
We know 180° = π
70° = 70 x (π/180) = 7π/18
Therefore, the third angle is 7π/18
Question 5:
Prove that 1° < 1c
Solution:
We know 180° = πc
1° = (π/180)c
1° = (22/7 × 180)c < 1c
Therefore, 1° < 1c