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If cos(2sin⁻¹x) = 1/9 then x =

a)
2/3
b)
±2/3
c)
-2/3
d)
none of these

Correct answer is option 'B'. Can you explain this answer?

Ref: https://edurev.in/question/504425/Ifcos-2sinminus1x-19then-x-a-23b-plusmn23c-23d-none-of-theseCorrect-answer-is-option-B-Can-you-

Given that,

cos ( 2 sin ̄¹ x ) = 1/9

Using Formula ,

{ cos 2θ = 1 − 2sin²θ }

We get

1 − 2 sin² ( sin ̄¹ x ) = 1/9

− 2 [ sin ( sin ̄¹ x ) ]² = 1/9 − 1

− 2 ( x )² = (1 − 9) / 9

− 2x² = − 8/9

→ x² = 8 / (9×2)

→ x² = 4/9

→ x = √ (4/9)

If cos(2sin^−1x) = 1/9 then x - Class 12

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FAQs on If cos(2sin^−1x) = 1/9 then x - Class 12

1. What is the value of x if cos(2sin^−1x) = 1/9?

Ans. To find the value of x, we can use the trigonometric identity cos(2θ) = 1 - 2sin^2(θ). Let's substitute sin^−1x with θ. Then, we have cos(2θ) = 1/9. Using the identity, we get 1 - 2sin^2(θ) = 1/9. Rearranging the equation, we have 2sin^2(θ) = 8/9. Dividing both sides by 2, we get sin^2(θ) = 4/9. Taking the square root, we have sin(θ) = ±2/3. Since sin(θ) represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle, the value of x can be ±2/3.

2. How can we solve the equation cos(2sin^−1x) = 1/9?

Ans. To solve the equation cos(2sin^−1x) = 1/9, we can use the double-angle identity for cosine, which states that cos(2θ) = 1 - 2sin^2(θ). We substitute sin^−1x with θ in the equation and get cos(2θ) = 1/9. By applying the identity, we have 1 - 2sin^2(θ) = 1/9. Rearranging the equation, we have 2sin^2(θ) = 8/9. Dividing both sides by 2, we get sin^2(θ) = 4/9. Taking the square root, we have sin(θ) = ±2/3. Hence, the solutions for x can be ±2/3.

3. What is the trigonometric identity used to solve the equation cos(2sin^−1x) = 1/9?

Ans. The trigonometric identity used to solve the equation cos(2sin^−1x) = 1/9 is the double-angle identity for cosine. It states that cos(2θ) = 1 - 2sin^2(θ). By substituting sin^−1x with θ in the equation, we can simplify and solve for x.

4. How do we simplify the equation cos(2sin^−1x) = 1/9 using trigonometric identities?

Ans. To simplify the equation cos(2sin^−1x) = 1/9, we can use the double-angle identity for cosine, which states that cos(2θ) = 1 - 2sin^2(θ). By substituting sin^−1x with θ in the equation, we get cos(2θ) = 1/9. Applying the identity, we have 1 - 2sin^2(θ) = 1/9. Rearranging the equation, we have 2sin^2(θ) = 8/9. Dividing both sides by 2, we obtain sin^2(θ) = 4/9. Taking the square root, we have sin(θ) = ±2/3, which determines the possible values of x.

5. How can we determine the values of x in the equation cos(2sin^−1x) = 1/9?

Ans. To determine the values of x in the equation cos(2sin^−1x) = 1/9, we can use the double-angle identity for cosine, which states that cos(2θ) = 1 - 2sin^2(θ). By substituting sin^−1x with θ in the equation, we simplify and solve for x. In this case, we find that sin(θ) = ±2/3. Since sin(θ) represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle, the possible values of x can be ±2/3.

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