Question

If sin theta = a and sin2theta= b, find the expression for cos theta in terms of a and b . Hence find a relation between a and b not involving theta

Answer 1

Answer:

Step-by-step explanation:

I am taking Theta to be 'A'

According to the Question,

⇒ Sin A = a

⇒ Sin 2A = b

We need to find Cos A in terms of a and b.

According to trigonometric identity,

⇒ Sin 2A = 2 × Sin A × Cos A

Substituting the values we get,

⇒ b = 2 × a × Cos A

⇒ Cos A = b / 2a

We know that,

⇒ Sin²A + Cos²A = 1   [ Trigonometric Identity ]

⇒ a² + ( b / 2a )² = 1

⇒ a² + b² / 4a² = 1

Taking LCM we get,

⇒ ( 4a⁴ + b² ) / 4a² = 1

⇒ 4a⁴ + b² = 4a²   [ Cross multiplying the denominator ]

⇒ 4a⁴ - 4a² = -b²

⇒ 4a² ( a² - 1 ) = -b²

⇒ 4a² ( 1 - a² ) = b²

⇒ √ ( 4a² ( 1 - a² )) = √ b²

⇒ 2a √ ( 1 - a² ) = b

This is the required relation.

Hope it helped !!

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Answer 2

• sinØ = a

• sin2Ø = b

Here.. Ø = theta

_____________ [ GIVEN ]

• We have to find the relation between a and b without involving theta.

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We know that..

→ sin2Ø = 2 sinØ cosØ

And we have given that sinØ = a

So,

→ sin2Ø = 2a cosØ

→ sin2Ø/2a = cosØ

→ cosØ = sin2Ø/2a

Also, sin2Ø = b

→ cosØ = b/2a _______ (eq 1)

______________________________

We also know that..

→ sin²Ø + cos²Ø = 1

Put value of sinØ and cosØ in above formula

→ (a)² + (b/2a)² = 1

→ a² + (b²/4a²) = 1

Take LCM

→ (4a⁴ + b²)/4a² = 1

→ 4a⁴ + b² = 4a²

→ b² = 4a² - 4a⁴

→ b² = 4a²(1 - a²)

______________________________

b² = 4a²(1 - a²) is the relation between a and b without involving theta (Ø).

___________ [ ANSWER ]

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