Problem 340 - Precalculus

Question

For the following exercises, use reference angles to evaluate the given expression. $$ \sec \frac{11 \pi}{3} $$

Step-by-Step Solution

Answer

The value of \( \sec \frac{11\pi}{3} \) is 2.

Step 1: Convert Angle to Degrees

First, convert the given radian angle to degrees to better understand where the angle lies in the unit circle. We know that \( \pi \text{ radians} = 180^\circ \). Therefore, \( \frac{11\pi}{3} \text{ radians} = \frac{11 \times 180}{3} = 660^\circ \).

Step 2: Find the Reference Angle

To find the reference angle of \(660^\circ\), we must first determine the equivalent angle within a full rotation (0 to 360 degrees). Subtract by 360 degrees: \(660^\circ - 360^\circ = 300^\circ \). So, \(300^\circ\) is equivalent to \(660^\circ\). The reference angle for \(300^\circ\) is \(360^\circ - 300^\circ = 60^\circ\).

Step 3: Evaluate the Secant Using Reference Angle

The secant function is the reciprocal of the cosine function. So, \( \sec(300^\circ) \) utilizes the reference angle \(60^\circ\). The cosine of \(60^\circ\) is \( \frac{1}{2} \). Therefore, the secant of \(300^\circ\) is the reciprocal: \( \sec(300^\circ) = \frac{1}{\frac{1}{2}} = 2 \). Since \(300^\circ\) is in the fourth quadrant where cosine is positive, \( \sec(660^\circ) = 2 \) as well.

Key Concepts

Reference Angles

When dealing with trigonometric problems, reference angles can be extremely helpful. A reference angle is the smallest angle between the terminal side of the given angle and the x-axis. This angle is always a positive acute angle (less than 90 degrees).

To find a reference angle:

• Identify which quadrant the terminal angle lies in.
• In the first quadrant, the reference angle is the same as the original angle.
• In the second quadrant, subtract the angle from 180 degrees.
• In the third quadrant, subtract 180 degrees from the angle.
• In the fourth quadrant, subtract the angle from 360 degrees.

Working with reference angles simplifies calculations because the trigonometric values for these angles are more commonly known or easily deduced. The understanding and application of reference angles allow for easier evaluation of trigonometric functions at larger or more complex angles.

Unit Circle

The unit circle is an essential tool in trigonometry, providing a geometric representation of all the angles and their corresponding sine and cosine values. The term "unit circle" refers to a circle with a radius of one unit, centered at the origin of the coordinate system.

In the unit circle:

• Any point on the circle can be represented by the coordinates (\(\cos(\theta), \sin(\theta)\)).
• The entire circle covers all angles from 0 degrees to 360 degrees (or 0 to 2\(\pi\) radians).
• Each quadrant of the circle reflects a different sign combination of sine and cosine.

For example, in Quadrant IV, cosine is positive, and sine is negative. Thus, knowing which quadrant an angle terminates helps determine the sign of its trigonometric function values, like secant, sine, and cosine.

Angle Conversion

Converting angles between degrees and radians is a routine task in trigonometry. This process allows for a uniform understanding when applying trigonometric functions, as different problems may present angles in either format.

The conversion from radians to degrees uses the relationship that \(\pi\) radians equals 180 degrees.

• To convert from radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).
• To convert from degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\).

Understanding these conversions is crucial, as it allows for correct interpretation and operation within the unit circle. For instance, in our example exercise, converting \(\frac{11\pi}{3}\) into degrees makes it clearer how this large angle relates to more familiar angles on a basic circle.

Secant Function

The secant function, usually abbreviated as sec, is one of the six fundamental trigonometric functions, and it is defined as the reciprocal of the cosine function.

• Mathematically, \(\sec(\theta) = \frac{1}{\cos(\theta)}\).
• It provides the ratio of the hypotenuse to the adjacent side in a right triangle.
• The secant function is undefined wherever cosine equals zero, which occurs at odd multiples of 90 degrees or \(\frac{\pi}{2}\) radians.

The exercise involved finding \(\sec(\frac{11\pi}{3})\). By utilizing the reference angle of 60 degrees (since the cosine of 60 degrees is \(\frac{1}{2}\)), it follows that the secant of 300 degrees or \(\frac{11\pi}{3}\) is 2, due to the positive cosine value in the fourth quadrant of the unit circle.