Problem 6 - Precalculus with Calculus Previews

Question

Find the indicated value without the use of a calculator. $$ \sec 7 \pi $$

Step-by-Step Solution

Answer

\(\sec 7\pi = -1\)

Step 1: Convert to Degrees

First, we will convert the given angle from radians to degrees. We know that \(\pi \, \text{radians} = 180^\circ\). Therefore, \(7\pi\, \text{radians} = 7 \times 180^\circ = 1260^\circ\).

Step 2: Simplify the Angle

Angles greater than 360° can be simplified by subtracting 360° until the angle is between 0° and 360°. So, \(1260^\circ - 3 \times 360^\circ = 180^\circ\). Thus, \(1260^\circ = 180^\circ \text{ in standard position}\).

Step 3: Determine \(\cos(180^\circ)\)

The cosine of 180° is well-known. From the unit circle, \(\cos(180^\circ) = -1\).

Step 4: Find \(\sec(180^\circ)\)

Secant is the reciprocal of cosine. Thus, \(\sec(180^\circ) = \frac{1}{\cos(180^\circ)} = \frac{1}{-1} = -1\).

Key Concepts

Radians to Degrees Conversion

Understanding how to convert between radians and degrees is essential in trigonometry, as each measure provides a different perspective on angles. Radians measure angles based on the radius of a circle, whereas degrees give a more visual estimate relative to the circle's entire rotation.

To convert an angle from radians to degrees, you use the conversion factor where \( \pi \) radians equals \( 180^\circ \). This means to change radians to degrees, you multiply by \( \frac{180}{\pi} \).

For example, if you have \( 7\pi \) radians like in our exercise, it would convert as follows:

• Multiply: \( 7 \pi \times \frac{180}{\pi} = 1260^\circ \).

Knowing this method allows you to work with angles in whichever measurement format is needed for the problem at hand.

Unit Circle

The unit circle is a crucial tool for understanding trigonometric functions. It is a circle with a radius of one, centered at the origin of the coordinate plane. Each point on the unit circle corresponds to an angle measured in radians or degrees.

Angles in the unit circle start at \(0^\circ\) (or \(0\) radians) on the positive x-axis and increase counterclockwise. A full circle is \(360^\circ\) or \(2\pi\) radians.

• This circle helps visualize and compute trigonometric values like sine, cosine, and tangent for specific angles.

For instance, key angles such as \(0^\circ\), \(90^\circ\), \(180^\circ\), and \(270^\circ\) are particularly important. In our example, we simplified \(1260^\circ\) by subtracting multiples of 360 to land at \(180^\circ\), finding it on the circle directly opposite \(0^\circ\). Such methods allow us to simplify and solve trigonometric functions accurately.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and hold true for any angle. They are immensely useful in simplifying expressions and solving problems.

A basic identity involves the cosine and secant functions. The secant function, \( \sec(\theta) \), is the reciprocal of the cosine function. This means \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Knowing this identity, you can determine secant values from given cosine values easily.

For \( 180^\circ \), located on the unit circle, \( \cos(180^\circ) = -1 \). Using the reciprocal identity, we find:

• \( \sec(180^\circ) = \frac{1}{-1} = -1 \).

Utilizing such identities not only simplifies calculations but also builds a deeper understanding of trigonometric relationships.