Question

Find the exact value of each expression. Do not use a calculator. $$ \sec \frac{11 \pi}{4} $$

Short answer

-\sqrt{2}

Step-by-step solution

Reduce the angle

First, reduce the angle \( \frac{11 \pi}{4} \) to a principal angle between \( 0 \) and \( 2\pi \). To do this, subtract multiples of \( 2\pi \) from \( \frac{11\pi}{4} \). Since \( 2\pi = \frac{8\pi}{4} \), we have \( \frac{11\pi}{4} - 2\pi = \frac{11\pi}{4} - \frac{8\pi}{4} = \frac{3\pi}{4} \). Hence, \( \frac{11\pi}{4} = \frac{3\pi}{4} \) in the principal range.

Identify the reference angle

The angle \( \frac{3\pi}{4} \) is in the second quadrant. To find the reference angle, subtract it from \( \pi \). The reference angle is \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \).

Determine the cosine of the reference angle

The reference angle is \( \frac{\pi}{4} \). In the second quadrant, cosine is negative. Thus, \( \cos \frac{3\pi}{4} = - \cos \frac{\pi}{4} \). Since \( \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \), it follows that \( \cos \frac{3\pi}{4} = -\frac{1}{\sqrt{2}} \).

Find the secant of the angle

Secant is the reciprocal of cosine. Therefore, \( \sec \frac{3\pi}{4} = \frac{1}{\cos \frac{3\pi}{4}} = \frac{1}{-\frac{1}{\sqrt{2}}} = -\sqrt{2} \).

Key concepts

unit circle

The unit circle is a circle centered at the origin (0,0) of a coordinate plane with a radius of 1. It is a powerful tool in trigonometry because it helps to define the sine, cosine, and tangent functions for all angles. < br > Every point on the unit circle corresponds to an \(x, y\) value, which can be interpreted as the coordinates \(\cos(\theta), \sin(\theta)\) for a given angle \(\theta\). < br > Some key points to remember about the unit circle:

• Each angle in the unit circle is measured in radians.
• The circle makes it easy to find the values of trigonometric functions at various angles.
• The values of sine and cosine range between -1 and 1, while tangent can take any real number.

Using the unit circle simplifies working with trigonometric functions and provides a visual understanding of how these functions behave at different angles.