Use reference angles to find the exact value of each expression. Do not use a calculator. sec 495°

Verified step by step guidance

1. First, recognize that the angle given is 495°, which is greater than 360°. To find its reference angle, reduce it by subtracting 360° to find an equivalent angle within one full rotation: \$495° - 360° = 135°$.
2. Next, determine the quadrant in which the angle 135° lies. Since 135° is between 90° and 180°, it lies in the second quadrant.
3. Recall that the secant function is the reciprocal of the cosine function: \(\sec \theta = \frac{1}{\cos \theta}\). The sign of secant depends on the sign of cosine in the quadrant of the angle.
4. Find the reference angle for 135°, which is the acute angle it makes with the x-axis. The reference angle \(\alpha\) is \$180° - 135° = 45°$.
5. Use the reference angle to find \(\cos 45°\), which is \(\frac{\sqrt{2}}{2}\). Since cosine is negative in the second quadrant, \(\cos 135° = -\frac{\sqrt{2}}{2}\). Therefore, \(\sec 135° = \frac{1}{\cos 135°} = -\sqrt{2}\). This value is the same for \(\sec 495°\) because they are coterminal angles.

Key Concepts

Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating angles greater than 90° to their acute counterparts, allowing the use of known exact values.

Secant Function and Its Relationship to Cosine

The secant function, sec(θ), is the reciprocal of the cosine function: sec(θ) = 1/cos(θ). Understanding this relationship is essential for finding exact values, especially when using reference angles to determine cosine values first.

Angle Reduction Using Coterminal Angles

Angles greater than 360° can be reduced by subtracting multiples of 360° to find a coterminal angle within the standard 0°–360° range. This simplification is crucial for applying reference angles and known trigonometric values.