Find the degree measure of θ if it exists. Do not use a calculator.
θ = arctan (-1)

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Recall that the function \( \arctan(x) \) gives the angle \( \theta \) whose tangent is \( x \), and its range is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) or \( (-90^\circ, 90^\circ) \).

Set up the equation \( \tan(\theta) = -1 \) and think about angles where the tangent value is \( 1 \) or \( -1 \).

Remember that \( \tan(45^\circ) = 1 \), so the angle with tangent \( -1 \) in the range of \( \arctan \) must be the negative of \( 45^\circ \), because tangent is negative in the fourth quadrant (or negative angles).

Therefore, the angle \( \theta \) satisfying \( \arctan(-1) \) is \( -45^\circ \).

Express the answer as the degree measure \( \theta = -45^\circ \), which lies within the principal range of the \( \arctan \) function.

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Inverse Trigonometric Functions

Inverse trigonometric functions, like arctan, return the angle whose trigonometric ratio equals a given value. For arctan(x), it gives the angle θ such that tan(θ) = x, typically within the principal range of -90° to 90°.

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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. Knowing common tangent values, such as tan(45°) = 1, helps identify angles when given a tangent value, including negative values indicating angles in specific quadrants.

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Angles can be measured in degrees and lie in four quadrants. Since arctan returns angles between -90° and 90°, a negative tangent value corresponds to an angle in the fourth quadrant (negative angle) or first quadrant (positive angle), guiding the correct angle selection.

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