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Three Square Problem

Three squares are placed in a row Connect the bottom left corner of each square to The top right corner of the third square These form three angles with the horizontal a,b,c 

What is a + b + c is equal to

easy-math-problems-that-look-hard sum of angles problems image

show that a + b + c = 90

Solution

using trigonometry and inverse trigonometry

Let the side of the square be one unit, then

c

{\color{DarkBlue} tan\;c=\frac{1}{1} } \\ {\color{DarkBlue} c=45\degree }

b

{ \color {DarkBlue}  tan\;b = \frac{1}{2} } \\ \\ \\ {\color {DarkBlue} b = tan^{-1}\frac{1}{2} }

a

{ \color {DarkBlue}  tan \; a =  \frac{1} {3} } \\ \\ \\ {\color {DarkBlue} a = tan^{-1}\frac{1}{3} }
a+b+c=tan^{-1}\frac{1}{3}+tan^{-1}\frac{1}{2}+45\degree

by using Inverse Trigonometric Function

{ \color{Red} tan ^ { -1 } a + tan ^ { -1 } b = tan ^{  - 1} \frac{a+b}{1-ab}, \; if \;the \;  value \; xy < 1}

tan^{-1}\frac{1}{3} + tan^ {- 1} \frac { 1 } {2} = tan^ {-1}  \frac { \frac {1} {3}+  \frac {1} {2} } { 1 - \frac {1} {3} \frac {1 } {2}} \\ tan^ {-1} \frac{ \frac{2+3} {6}} { \frac {6-1} {6} }
= tan ^  { -1 }  \frac {  \frac {5} {6} } {  \frac {5} {6} }=tan^ {-1} 1 = 45 \degree
 {\color{Teal} a + b  = 45 \degree}  \; and \; {\color{Teal} c=45 \degree}

Sum of three angles is

 { \color {Teal} a + b + c  = 90  \degree}  

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