Principle of Mathematical Induction

Principle of Mathematical Induction

Question

The angles of a triangle are in A.P. and the number of degrees in the least angle is to the number of radians in the greatest angle is 60 : straight pi. Find the angles in degrees.

Answer

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#6 {main}</pre> be the three angles of a triangle in A.P. with d>0
Since the sum of the angles of a triangle is 180 degree
∴         left parenthesis straight a minus straight d right parenthesis space plus space straight a space plus space left parenthesis straight a space plus space straight d right parenthesis space equals space 180 space rightwards double arrow space 3 straight a space equals space 180 space rightwards double arrow space straight a space equals space 60
Now, number of degrees in the least angle = (60 - d)
Number of degrees in the greatest angle  = (60 + d)

rightwards double arrow Number of radians in the greatest angle = open square brackets fraction numerator left parenthesis 60 plus straight d right parenthesis straight pi over denominator 180 end fraction close square brackets
According to the given condition,
left parenthesis 60 minus straight d right parenthesis space colon space fraction numerator left parenthesis 60 plus straight d right parenthesis straight pi over denominator 180 end fraction space space equals space 60 space colon space straight pi space space space rightwards double arrow space space space space fraction numerator left parenthesis 60 minus straight d right parenthesis 180 over denominator left parenthesis 60 plus straight d right parenthesis straight pi end fraction space equals space 60 over straight pi
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#6 {main}</pre>     rightwards double arrow space space space space 180 space minus space 3 straight d space equals space 60 space plus space straight d   rightwards double arrow      4d = 120     rightwards double arrow d = 30
Hence, the angles of a triangle in degrees are:
space space space space space space left parenthesis 60 minus 30 right parenthesis to the power of ring operator comma space 60 degree comma space space left parenthesis 60 plus 30 right parenthesis degree     or      30 degree comma space 60 degree comma space 90 degree

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