Question

. If x = a sinø and y = b cos ø, what is the value of b²x² + a²y²??​

Answer 1

$\bigstar\:\underline{\underline{\large{\purple{\mathcal{GIVEN}}}}}$

• x = a sin ∅   ... eqn (1)
• y = b cos ∅   ....eqn (2)

$\bigstar\:\underline{\underline{\large{\purple{\mathcal{TO\:FIND}}}}}$

• b² x² + a² y² = ?

$\bigstar\:\underline{\underline{\large{\purple{\mathcal{TRIGONOMETRIC \;\:IDENTITIES\:\; USED}}}}}$

• sin²θ + cos²θ = 1

$\bigstar\:\underline{\underline{\large{\purple{\mathcal{SOLUTION}}}}}$

▶  b² x² + a² y²

using eqn (1) and (2)

▶ b² ( a sin∅ )² + a² ( b cos∅ )²

▶ b² a² sin²∅ + a² b² cos²∅

taking   b²a²  common

▶ b² a² ( sin²∅ + cos²∅ )

using trigonometric identity

sin²θ + cos²θ = 1

▶ b² a² ( 1 )

▶ ( ab )²       ( Ans. )

$\bigstar\:\underline{\underline{\large{\purple{\mathcal{MORE\:\;TRIGONOMETRIC\:\;IDENTITIES\:, RATIOS}}}}}$

⇰ 1 + tan²θ = sec²θ

⇰ 1 + cot²θ = cosec²θ

⇰ cosec θ = 1 / sinθ

⇰ sec θ = 1 / cosθ

⇰ tanθ = sinθ / cosθ

⇰ cotθ = cosθ / sinθ

⇰ sin ( 90 - θ ) = cos θ

⇰ cos ( 90 - θ ) = sin θ

⇰ cosec ( 90 - θ ) = sec θ

⇰ sec ( 90 - θ ) = cosec θ

⇰ tan ( 90 - θ ) = cot θ

⇰ cot ( 90 - θ ) = tan θ