 To convert the trigonometric angle units from circular to sexagesimal, you need to know the relationship between them. The unit to measure the angle in a circular system is radian and in the sexagesimal system is degrees, minutes and seconds. Have a look at the following sections to get the complete details on how to convert from circular to the sexagesimal system along with the worked-out questions.

## Conversion from Circular to Sexagesimal System

In trigonometry, the circular system, sexagesimal system are the two different systems that are helpful to measure the angle and those units. The unit of the circular system is radian and the sexagesimal system is degrees, minutes and seconds. Here we are providing the simple steps to convert radians to degree minutes and seconds.

• Get the circular measure of the angle.
• Substitute 180 degrees in the place of π in the angle.
• Solve it to find angle measure in degrees.
• Now, multiply the decimal degrees by 60 to get minutes.
• Again multiply the remaining decimals by 60 to get seconds.

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### Problems on Converting Circular to Sexagesimal System

Problem 1:
Convert $$\frac { 5π }{ 12 }$$ into sexagesimal.

Solution:
We know, πc = 180°
So, $$\frac { 5π }{ 12 }$$ = $$\frac { 5 x 180 }{ 12 }$$
= 75°

Problem 2:
In a right-angled triangle the difference between two acute angles is $$\frac { 2π }{ 5 }$$. Express these two angles in terms of radian and degree.

Solution:
Let the two acute angles by x and y
x + y = $$\frac { π }{ 2 }$$, x – y = $$\frac { 2π }{ 5 }$$
2x = $$\frac { π }{ 2 }$$ + $$\frac { 2π }{ 5 }$$
= $$\frac { 9π }{ 10 }$$
x = $$\frac { 9π }{ 20 }$$
y = $$\frac { π }{ 2 }$$ – $$\frac { 9π }{ 20 }$$
= $$\frac { π }{ 20 }$$
As π = 180°
x = $$\frac { 9 x 180° }{ 20 }$$ = 81°
y = $$\frac { 180° }{ 20 }$$ = 9°
The angles in radians are $$\frac { 9π }{ 20 }$$, $$\frac { π }{ 20 }$$ and in degrees are 81°, 9°.

Problem 3:
ABC is an equilateral triangle where AD is a line segment that joins the vertex A to the mid point of BC. What is the sexagesimal measure of ∠BAD?

Solution:
Given that,
∆ABC is equilateral
Therefore, ∠BAC = $$\frac { π }{ 3}$$
We also know that the median of an equilateral triangle bisects the corresponding vertice angle. Therefore, ∠BAD = $$\frac { π }{ 6}$$
Therefore, the sexagesimal measure of ∠BAD = $$\frac { 180 }{ 6}$$ = 30°

Problem 4:
The circular measure of an angle is $$\frac { π }{ 7}$$; find its value in sexagesimal systems.

Solution:
Given circular angle is $$\frac { π }{ 7}$$
We know, πc = 180°
$$\frac { π }{ 7}$$ = $$\frac { 180 }{ 7}$$
= 25.7142
= 25° + 0.7142°
[Now we will convert 0.7142° to minute.]
= 25° + (0.7142 x 60)’ [since 1° = 60’]
= 25° + 42.852′
= 25° + 42′ + (0.852)’
[Now we will convert 0.852′ to seconds.]
= 25° + 42′ + (0.852 x 60)” [since 1′ = 60″]
= 25° + 42′ +51″
Therefore, the sexagesimal measure of angle $$\frac { π }{ 7}$$ is 25° 42′ 51″.

Problem 5:
The two angles of a triangle are $$\frac { 3π }{ 10 }$$, $$\frac { π }{ 5 }$$. Find the values of all three angles in sexagesimal measure.

Solution:
In ∆ABC, ∠ABC = $$\frac { 3π }{ 10 }$$ and ∠ACB = $$\frac { π }{ 5 }$$; ∠BAC = ?
The sum of internal angles of a triangle is π
So, ∠BAC = π – ($$\frac { 3π }{ 10 }$$ + $$\frac { π }{ 5 }$$)
= π – ($$\frac { π }{ 2 }$$)
= $$\frac { π }{ 2 }$$
∠ABC = $$\frac { 3π }{ 10 }$$ = $$\frac { 3 x 180 }{ 10 }$$ = 54°
∠ACB = $$\frac { π }{ 5 }$$ = $$\frac { 180 }{ 5 }$$ = 36°
∠BAC = $$\frac { π }{ 2 }$$ = $$\frac { 180 }{ 2 }$$ = 90°
Therefore, the angle measures of triangle in sexagesimal are 90°, 36° and 54°.