Evaluate the expression sin^(-1)(x) + sin^(-1)(√(1-x^2)).

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Question: Evaluate the expression sin^(-1)(x) + sin^(-1)(√(1-x^2)).

Options:

Correct Answer: π/2

Solution:

Using the identity sin^(-1)(x) + sin^(-1)(√(1-x^2)) = π/2 for x in [0, 1], the value is π/2.

Evaluate the expression sin^(-1)(x) + sin^(-1)(√(1-x^2)).

Evaluate the expression sin^(-1)(x) + sin^(-1)(√(1-x^2)).

Step 1: Understand the notation sin^(-1)(x). This represents the inverse sine function, also known as arcsin(x).
Step 2: Recognize that the expression sin^(-1)(x) + sin^(-1)(√(1-x^2)) involves two arcsin functions.
Step 3: Recall the identity that states sin^(-1)(x) + sin^(-1)(y) = π/2 when x and y are related by the equation x^2 + y^2 = 1.
Step 4: In our case, let y = √(1-x^2). Notice that x^2 + (√(1-x^2))^2 = x^2 + (1-x^2) = 1. This means the identity applies.
Step 5: Therefore, we can conclude that sin^(-1)(x) + sin^(-1)(√(1-x^2)) = π/2.
Step 6: Since this is true for x in the range [0, 1], we can state the final result.

Inverse Trigonometric Functions – Understanding the properties and identities of inverse sine functions.
Trigonometric Identities – Applying the identity that relates the sum of inverse sine functions to π/2.
Domain of Functions – Recognizing the valid input range for the inverse sine function.