Question

Consider ∆ACB, right - angled at C, in which AB = 29 Units, BC = 21 units and angle ABC = θ. Determine the values of –

(1) cos²θ + sin²θ
(2) cos²θ - sin²θ​

Answer 1

★ Given :

• AB = 29 Units,
• BC = 21 units,
• ∠ABC = θ.

★ To Find :

• Determine the values of –

1. (1) cos²θ + sin²θ
2. (2) cos²θ - sin²θ

★ Solution :

In ∆ACB, we have :

${ \rm{ \implies{AC = \sqrt{AB^{2} - BC^{2} } = \sqrt{(29)^{2} - (21) ^{2}}}} }$

${ \rm{ \implies{ \sqrt{(29 - 21)(29 + 21)} = \sqrt{(8)(50)} }}}$

${ \rm{ \implies{ \sqrt{400} = 20 \: units}}}$

${ \rm{ \implies{So{ \boxed{ \rm{ \blue{ \: \sin θ = \frac{AC}{AB} = \frac{20}{29} \: , \: \cos θ = \frac{BC}{AB} = \frac{21}{29} }}} }}}}$

${\rm{(1) \cos^{2} θ + \sin ^{2} θ}}$

$\begin{gathered} \\ \rm \implies \left( \dfrac{20}{29} \right)^{2} + \left( \dfrac{21}{29} \right)^{2} \\ \end{gathered}$

${\sf{ \implies{ \frac{20^{2} + 21 ^{2} }{29 ^{2}} = \frac{400 + 441}{841} = { \boxed{ \sf{ \red{{1}}}}}}}}$

${\rm{(1) \cos^{2} θ - \sin ^{2} θ}}$

$\begin{gathered} \\ \rm \implies \left( \dfrac{21}{29} \right)^{2} - \left( \dfrac{20}{29} \right)^{2} \\ \end{gathered}$

$\implies{ \sf{ \frac{(21 + 20)(21 - 20)}{29 ^{2} } = { \boxed{ \sf{ \red{ \frac{41}{841} }}}}}}$

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