Find the exact value of each real number y if it exists. Do not use a calculator.
y = arctan 0

Verified step by step guidance

Recall that the function \( y = \arctan(x) \) gives the angle \( y \) whose tangent is \( x \), and the range of \( \arctan \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

Set up the equation based on the problem: \( y = \arctan(0) \) means \( \tan(y) = 0 \).

Identify all angles \( y \) within the range \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) where \( \tan(y) = 0 \).

Recall that \( \tan(0) = 0 \), and since \( 0 \) is within the range of \( \arctan \), this is the principal value.

Conclude that the exact value of \( y \) is \( 0 \) because it satisfies \( \tan(y) = 0 \) within the principal range.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definition of the Arctangent Function

The arctangent function, denoted as arctan or tan⁻¹, is the inverse of the tangent function restricted to the interval (-π/2, π/2). It returns the angle whose tangent is the given number, providing a unique output for each real input.

Tangent Function and Its Values

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Key values include tan(0) = 0, which means the angle whose tangent is zero is 0 radians (or 0 degrees). This helps identify the exact value of arctan(0).

Exact Values Without a Calculator

Certain trigonometric values are well-known and can be determined exactly without a calculator, such as arctan(0) = 0. Recognizing these standard angles and their trigonometric ratios is essential for solving problems involving inverse trig functions precisely.

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