Find the exact value of each expression. See Example 3. sec(-495°)

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Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). So, to find \(\sec(-495^\circ)\), we first need to find \(\cos(-495^\circ)\).

Use the even-odd property of cosine: \(\cos(-\theta) = \cos \theta\). Therefore, \(\cos(-495^\circ) = \cos(495^\circ)\).

Reduce the angle \(495^\circ\) to an equivalent angle between \(0^\circ\) and \(360^\circ\) by subtracting \(360^\circ\): \(495^\circ - 360^\circ = 135^\circ\). So, \(\cos(495^\circ) = \cos(135^\circ)\).

Recall the value of \(\cos(135^\circ)\). Since \(135^\circ\) is in the second quadrant where cosine is negative, and \(135^\circ = 180^\circ - 45^\circ\), use the identity \(\cos(180^\circ - \theta) = -\cos \theta\) to find \(\cos(135^\circ) = -\cos(45^\circ)\).

Use the known exact value \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\). Substitute back to find \(\cos(135^\circ) = -\frac{\sqrt{2}}{2}\). Finally, calculate \(\sec(-495^\circ) = \frac{1}{\cos(-495^\circ)} = \frac{1}{-\frac{\sqrt{2}}{2}}\).

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주요 개념

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Angle Reduction Using Coterminal Angles

Angles differing by full rotations (360°) share the same trigonometric values. To simplify sec(-495°), add or subtract multiples of 360° to find a coterminal angle between 0° and 360°, making evaluation easier.

Definition of Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function: sec(θ) = 1/cos(θ). Knowing this relationship allows you to find secant values once the cosine of the angle is determined.

Evaluating Trigonometric Functions at Standard Angles

Certain angles have well-known exact trigonometric values (e.g., 0°, 30°, 45°, 60°, 90°). After reducing the angle to a standard position, use these known values to find the exact value of sec(θ).