Find the value of sin^(-1)(√(1 - cos^2(θ))).

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Question: Find the value of sin^(-1)(√(1 - cos^2(θ))).

Options:

Correct Answer: θ

Solution:

Since sin^2(θ) = 1 - cos^2(θ), we have sin^(-1)(√(1 - cos^2(θ))) = sin^(-1)(sin(θ)) = θ.

Find the value of sin^(-1)(√(1 - cos^2(θ))).

Find the value of sin^(-1)(√(1 - cos^2(θ))).

Step 1: Recall the Pythagorean identity which states that sin^2(θ) + cos^2(θ) = 1.
Step 2: Rearrange the identity to find sin^2(θ): sin^2(θ) = 1 - cos^2(θ).
Step 3: Substitute this into the expression: sin^(-1)(√(1 - cos^2(θ))) becomes sin^(-1)(√(sin^2(θ))).
Step 4: Since √(sin^2(θ)) is equal to |sin(θ)|, we have sin^(-1)(|sin(θ)|).
Step 5: The inverse sine function, sin^(-1)(x), gives the angle whose sine is x. Therefore, sin^(-1)(|sin(θ)|) = θ if θ is in the range of the inverse sine function (which is -π/2 to π/2).
Step 6: Thus, we conclude that sin^(-1)(√(1 - cos^2(θ))) = θ.

Inverse Trigonometric Functions – Understanding the relationship between sine, cosine, and their inverse functions.
Pythagorean Identity – Using the identity sin^2(θ) + cos^2(θ) = 1 to simplify expressions.