Problem 29 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \sec \frac{17 \pi}{3} $$

Step-by-Step Solution

Answer

The exact value of \( \sec \frac{17\pi}{3} \) is 2.

Step 1: Determine the reference angle

First, we need to convert the angle \( \frac{17\pi}{3} \) to a reference angle within the first revolution. A full circle is \( 2\pi \) radians. Divide \( \frac{17\pi}{3} \) by \( 2\pi \) to determine how many full revolutions are within this angle. We have \( \frac{17\pi}{3} \div 2\pi = \frac{17}{6} \) revolutions. The integer part is \( 2 \) full revolutions (since \( \left\lfloor \frac{17}{6} \right\rfloor = 2 \)). Subtracting \( 2 \times 6\pi \) gives a reference angle of \( \frac{17\pi}{3} - 4\pi = \frac{5\pi}{3} \).

Step 2: Determine the angle's standard position

The reference angle \( \frac{5\pi}{3} \) is greater than \( 2\pi \). Since \( \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{\pi}{3} = 2\pi - \frac{\pi}{3} \), this indicates the angle is in the fourth quadrant with reference angle \( \frac{\pi}{3} \).

Step 3: Calculate the Secant Function

In the fourth quadrant, the cosine of the angle is positive. Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \). The reference angle is \( \frac{\pi}{3} \), where \( \cos \frac{\pi}{3} = \frac{1}{2} \). Thus, \( \sec \frac{\pi}{3} = \frac{1}{\frac{1}{2}} = 2 \). Therefore, \( \sec \frac{17\pi}{3} = 2 \) because the secant function has the same sign as cosine in the fourth quadrant.

Key Concepts

Secant Function

The secant function, denoted as \( \sec(\theta) \), is one of the reciprocals of the basic trigonometric functions. In essence, it is the reciprocal of the cosine function. This simply means:

• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• The secant function is undefined where the cosine function is zero.
• It is periodic, repeating its values every \(2\pi\) radians, similar to other trigonometric functions.

The secant function is important when calculating angles that are part of geometric problems or periodic motion. Given the properties of cosine, the secant function will change its sign depending on the quadrant in which the angle \(\theta\) lies. This is crucial when identifying the angle’s position in relation to the x-axis.

Reference Angle

A reference angle is a way to simplify the evaluation of trigonometric functions for angles in any quadrant. It is the acute angle that a main angle makes with the x-axis, irrespective of which quadrant the main angle lies in. To find the reference angle:

• Subtract the largest multiple of \(\pi\) that keeps the resulting angle between \(0\) and \(\pi\).
• Ensure it is an acute angle (less than \(\pi/2\)).

In the context of the exercise, the reference angle for \(\frac{17\pi}{3}\) was reduced to \(\frac{\pi}{3}\) after accounting for the revolutions around the circle. This simplification allows us to use known trigonometric values to easily solve the problem.

Radians in a Circle

Radian is a measure of angle based on the radius of a circle. When an angle is measured in radians, it simply expresses the ratio of the arc length to the radius of the circle. Key points include:

• A full circle encompasses \(2\pi\) radians.
• Half a circle, or a straight angle, is \(\pi\) radians.
• When dealing with angles like \(\frac{17\pi}{3}\), we calculate how many times it circles around the circle by dividing it by \(2\pi\).

Understanding radians helps simplify the task of converting and comparing angles, as we can relate angles directly to their movement around a circle without needing to use degrees.

Fourth Quadrant

The fourth quadrant is one of the four sections into which the plane is divided by the x and y axes. Any angle in standard position that ends between \(270^\circ\) and \(360^\circ\) or \(\frac{3\pi}{2}\) and \(2\pi\) lies in this quadrant. Distinct features of the fourth quadrant include:

• Sine is negative, but cosine and secant are positive.
• Angles in this quadrant appear just before completing a full revolution.
• It is the quadrant where trigonometric functions change their sign as they generally return to zero around \(360^\circ\) or \(2\pi\).

In our exercise, \(\frac{17\pi}{3}\) reduces to an angle in the fourth quadrant, which is significant as it helps us determine the sign of the secant function. Knowing that cosine is positive in the fourth quadrant confirms that the secant function will also yield a positive value.