Workedout 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° = π^{c}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 anticlockwise 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
Measurement of Trigonometric Angles
Relation between Sexagesimal and Circular
Conversion from Sexagesimal to Circular System
Conversion from Circular to Sexagesimal System
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