If cos^(-1)(x) = θ, then what is the value of sin(θ)?

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Question: If cos^(-1)(x) = θ, then what is the value of sin(θ)?

Options:

Correct Answer: √(1-x^2)

Solution:

From cos^(-1)(x) = θ, we have cos(θ) = x. Therefore, sin(θ) = √(1 - cos^2(θ)) = √(1 - x^2).

If cos^(-1)(x) = θ, then what is the value of sin(θ)?

If cos^(-1)(x) = θ, then what is the value of sin(θ)?

Step 1: Start with the equation cos^(-1)(x) = θ. This means that θ is the angle whose cosine is x.
Step 2: From the definition of cosine, we know that cos(θ) = x.
Step 3: Use the Pythagorean identity, which states that sin^2(θ) + cos^2(θ) = 1.
Step 4: Substitute cos(θ) with x in the identity: sin^2(θ) + x^2 = 1.
Step 5: Rearrange the equation to find sin^2(θ): sin^2(θ) = 1 - x^2.
Step 6: Take the square root of both sides to find sin(θ): sin(θ) = √(1 - x^2).

Inverse Trigonometric Functions – Understanding the relationship between inverse cosine and sine functions.
Pythagorean Identity – Using the identity sin²(θ) + cos²(θ) = 1 to derive sin(θ) from cos(θ).