Question

1) tanΘ = a/b, Prove that. a sinΘ + b sinΘ = √a² + b²​

Answer 1

Correct Question:-

If tan θ = a/b , then prove that:

a sin θ + b cos θ = √a² + b²

Answer:-

Given:

tan θ = a/b

We know that,

tan θ = Opposite side/Adjacent side

So,

Opposite side / adjacent side = a / b

This implies ,

• Opposite side = a

• Adjacent side = b

Using Pythagoras Theorem,

(Hypotenuse)² = (Adjacent side)² + (Opposite side)²

So,

(Hypotenuse)² = b² + a²

⟶ Hypotenuse = √a² + b²

Now,

we know,

sin θ = Opposite side/Hypotenuse.

So,

★ sin θ = a / √a² + b²

Similarly,

★ cos θ = b / √a² + b²

Now,

We have to prove:

a sin θ + b cos θ = √a² + b²

⟶ a [ a / √a² + b² ] + b [ b / √a² + b² ] = √a² + b²

⟶ (a² + b²) / √a² + b² = √a² + b²

⟶ (√a² + b²)(√a² + b²) / √a² + b² = √a² + b²

[ Since, a = √a × √a ]

⟶ √a² + b² = √a² + b²

Hence, Proved.