Consider A C B right angled at C in which AB = 29 units, BC = 21 units and A B C= . Determine the values of 1. ^2 + ^2 2. ^2 - ^2

In, A C B we have A B ^ 2 = A C ^ 2 + B C ^ 2 A C = A B ^ 2 - B C ^ 2 = 29 ^ 2 - 21 ^ 2 = ( 29 + 21 ) ( 29 - 21 ) = 400 = 20 units = A C A B = 20 29 and = B C A B = 21 29 1. Using the values of and, we get ^ 2 + ^ 2 = ( 21 29 ) ^ 2 + ( 20 29 ) ^ 2 = 441 + 400 841 = 1 2. Using the values of and, we obtain ^2 - ^2 = (21 29 )^2- (20 29 )^2=21^2-20^2 29^2 =(21+20)(21-20) 841 =41 841

Consider A C B right angled at C in which AB = 29 units, BC = 21 …

Download App

Question

Consider $ \triangle A C B$ right angled at $C$ in which $AB = 29$ units, $BC = 21$ units and $ \angle A B C$= $\theta$. Determine the values of

$ \cos ^2 \theta+\sin ^2 \theta $
$ \cos ^2 \theta-\sin ^2 \theta$

✓

Answer

In, $ \Delta A C B$ we have
$ A B ^ { 2 } = A C ^ { 2 } + B C ^ { 2 }$
$ \Rightarrow \quad A C = \sqrt { A B ^ { 2 } - B C ^ { 2 } } = \sqrt { 29 ^ { 2 } - 21 ^ { 2 } } = \sqrt { ( 29 + 21 ) ( 29 - 21 ) } = \sqrt { 400 } = 20$ units
$ \therefore \quad \sin \theta = \frac { A C } { A B } = \frac { 20 } { 29 }$and $ \cos \theta = \frac { B C } { A B } = \frac { 21 } { 29 }$

Using the values of $ \sin \theta$ and,$ \cos \theta$ we get $ \cos ^ { 2 } \theta + \sin ^ { 2 } \theta = \left( \frac { 21 } { 29 } \right) ^ { 2 } + \left( \frac { 20 } { 29 } \right) ^ { 2 }$
$ = \frac { 441 + 400 } { 841 } = 1$
Using the values of $ \sin \theta$and,$ \cos \theta$ we obtain
$\cos^2\theta-\sin^2\theta=\left(\frac{21}{29}\right)^2-\left(\frac{20}{29}\right)^2=\frac{21^2-20^2}{29^2}=\frac{(21+20)(21-20)}{841}=\frac{41}{841}$