Question

Q.8 The sum of two angles is 5π and their
difference is 60°. Find their measures in
degree.
​

Answer 1

Given :

• Sum of the two angles = 5π

• Difference of the two angles = 60°

To find :

The measures of the angles.

Solution :

First us find the measure of 5π in degrees.

We know the formula :-

$:\implies \bf{\dfrac{D}{90^{\circ}} = \dfrac{R}{\dfrac{\pi}{2}}}$

Where :-

• D = Degrees
• R = Radians

Using the above formula and substituting the values in it , we get :-

$:\implies \bf{\dfrac{D}{90^{\circ}} = \dfrac{5\pi}{\dfrac{\pi}{2}}}$

$:\implies \bf{\dfrac{D}{90^{\circ}} = \dfrac{2 \times 5\pi}{\pi}}$

$:\implies \bf{\dfrac{D}{90^{\circ}} = \dfrac{10\pi}{\pi}}$

$:\implies \bf{\dfrac{D}{90^{\circ}} = \dfrac{10\not{\pi}}{\not{\pi}}}$

$:\implies \bf{\dfrac{D}{90^{\circ}} = 10}$

$:\implies \bf{D = 90^{\circ} \times 10}$

$:\implies \bf{D = 900^{\circ}}$

$\underline{\therefore \bf{D = 900^{\circ}}}$

Hence, the measure of 5π in degrees is 900°.

Now,

To find the measure of two angles :

Let the first angle be x and second angle be y.

So , According to the Question ,

we get the first as :-

$\underline{\boxed{:\implies \bf{x + y = 900^{\circ}}}}$

And the second Equation as :-

$\underline{\boxed{:\implies \bf{x - y = 60^{\circ}}}}$

Now , by subtracting Equation (ii) from Equation (i) , we get :-

$:\implies \bf{(x + y) - (x - y) = 900^{\circ} - 60^{\circ}}$

$:\implies \bf{x + y - x + y = 900^{\circ} - 60^{\circ}}$

$:\implies \bf{\not{x} + y - \not{x} + y = 900^{\circ} - 60^{\circ}}$

$:\implies \bf{y + y = 900^{\circ} - 60^{\circ}}$

$:\implies \bf{2y = 840^{\circ}}$

$:\implies \bf{y = \dfrac{840^{\circ}}{2}}$

$:\implies \bf{y = 420^{\circ}}$

$\underline{\therefore \bf{y = 420^{\circ}}}$

Hence, the second angle is 420°.

Now , Substituting the value of y in the first Equation , we get :-

$:\implies \bf{x + y = 900^{\circ}}$

$:\implies \bf{x + 420^{\circ} = 900^{\circ}}$

$:\implies \bf{x = - 420^{\circ} + 900^{\circ}}$

$:\implies \bf{x = 480^{\circ}}$

$\underline{\therefore \bf{x = 480^{\circ}}}$

Hence, the first angle is 480°.

Thus, the two angles are 480° and 420°.