In this topic convert into radian, we will learn how to convert the other units into radian units. The problems are based on changing the different measuring units to radian units.

Worked-out problems to convert into Radian:

1. The angles of a triangle are in A. P. If the greatest and the least are in the ratio 5 : 2, find the angles of the triangle in radian.

Solution:

Let (a - d), a and (a + d) radians (which are in A. P.) be the angles of the triangle where a> 0 and d > 0. Then,
a - d + a + a + d = π, (Since, the sum of the three angles of a triangle = 180° = π radian)

or, 3a = π

or, a =  π/3.

By problem, we have,

(a + d)/(a – d) = 5/2

or, 5(a – d) = 2(a + d)

or, 5a - 5d = 2a + 2d.

or, 5a – 2a = 2d + 5d

or, 3a = 7d

or, 7d = 3a

or, d = (3/7)a

or, d = (3/7)  × (π/3)

or, d = π/7

Therefore, the required angles of the triangle are (π/3- π/7), π/3 and (π/3 + π/7) radians

i.e.,  4π/21, π/3 and 10π/21 radians.

2. The sum of number of degrees, minutes and seconds of an angle is 43932; find the angle in circular system.

Solution:

Let, D, M and S be the measure of the angle in degree, minute and second units respectively. Then by question,

D  + M + S = 43932   ………….. (A)

Now, the value of the angle in degree unit = D

Therefore, the value of the angle in minute unit  = 60D [Since, 1° = 60’]

And the value of the angle in second unit  = 60 × 60 D

= 3600D [Since, 1’ = 60”]

By question, M = 60D and S = 3600D.

Now, putting the values of M and S in (A)

We get, D + 60D + 3600D = 43932

Or, 3661D = 43932

or, D =  43932/3661

or, D = 12.

Hence, the angle measures 12°.

Therefore, the value of the angle in circular system = 12 πc / 180 [As, 180° = πc] = πc/15.

Measurement of Angles

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