Question

Consider $△ACB$, right angled at $C$, in which $AB=29$ units, $BC=21$ units $\angle ABC=\theta$. Determine the value of
(i) ${\cos }^2\theta +{\sin }^2\theta$
(ii) ${\cos }^2\theta -{\sin }^2\theta$

Answer 1

By Pythagoras theorem,

$A{B}^2=A{C}^2+B{C}^2$

$\Rightarrow A{C}^2=A{B}^2-B{C}^2$

$\Rightarrow A{C}^2={29}^2-{21}^2$

$=(29+21)(29-21)=50\times 8$

$\Rightarrow A{C}^2=25\times 16$

$\Rightarrow AC=5\times 4=20$ units

$\Rightarrow \cos \theta =\frac{BC}{AB}=\frac{21}{29}$,

$\sin \theta =\frac{AC}{AB}=\frac{20}{29}$

$\Rightarrow \sin 2\theta +{\cos }^2\theta =\frac{{21}^2+{20}^2}{{29}^2}=\frac{{29}^2}{{29}^2}=1$

$\Rightarrow {\cos }^2\theta -{\sin }^2\theta =\frac{{21}^2-{20}^2}{{29}^2}=\frac{41}{841}$.