Problem 29 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The exact value is 2.

Step 1: Convert angle to an equivalent angle in standard position

First, we need to express \( \frac{17\pi}{3} \) in an equivalent angle between \( 0 \) and \( 2\pi \). To do this, subtract multiples of \( 2\pi \) from \( \frac{17\pi}{3} \) until the angle is in the desired range:\[ \frac{17\pi}{3} - 2\pi \times 2 = \frac{17\pi}{3} - \frac{12\pi}{3} = \frac{5\pi}{3} \]Thus, \( \frac{17\pi}{3} \) is equivalent to \( \frac{5\pi}{3} \) in standard position.

Step 2: Determine the related reference angle

The reference angle for \( \frac{5\pi}{3} \) can be found by realizing this angle is in the fourth quadrant, where angles are expressed generally as \( 2\pi - \theta \). The reference angle is:\[ 2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3} \]

Step 3: Calculate the cosine of the reference angle

The cosine of the reference angle \( \frac{\pi}{3} \) is known from trigonometric ratios. For \( \frac{\pi}{3} \), we have\[ \cos \frac{\pi}{3} = \frac{1}{2} \]

Step 4: Determine the sign of cosine in the fourth quadrant

In the fourth quadrant, the cosine function is positive. Therefore, \( \cos \frac{5\pi}{3} = \frac{1}{2} \).

Step 5: Find the secant of the angle

Secant is the reciprocal of cosine. Thus, we find \( \sec \frac{5\pi}{3} \) by taking the reciprocal of \( \cos \frac{5\pi}{3} \):\[ \sec \frac{5\pi}{3} = \frac{1}{\cos \frac{5\pi}{3}} = \frac{1}{\frac{1}{2}} = 2 \]

Key Concepts

angle conversion

Converting angles is an essential step in solving many trigonometric problems. Sometimes, you might encounter angles expressed in radians beyond the typical range from 0 to \(2\pi\). To handle this, we reduce them to their equivalent angles within this range. For instance, let's look at \(\frac{17\pi}{3}\).
To convert, we subtract whole multiples of \(2\pi\) (since one full circle in trigonometry is \(2\pi\) radians) until the result falls between 0 and \(2\pi\). Here's the calculation:

• First, compute a suitable multiple of \(2\pi\) within the denominator's framework: \(2\pi \times 2 = \frac{12\pi}{3}\).
• Subtract from the original angle: \(\frac{17\pi}{3} - \frac{12\pi}{3} = \frac{5\pi}{3}\).

Thus, \(\frac{17\pi}{3}\) corresponds to \(\frac{5\pi}{3}\), an angle in standard position. This approach simplifies the trigonometric analysis in a specific quadrant.

reference angle

Once you've situated your angle in a standard position, the next step is often to identify the reference angle. Reference angles help simplify the evaluation of trigonometric functions by reducing it to familiar angles in the first quadrant. These angles are always less than \(\pi / 2\).
For \(\frac{5\pi}{3}\), which lies in the fourth quadrant, the reference angle \(\theta_{ref}\) is found using the formula:

• \(\theta_{ref} = 2\pi - \theta\)

Calculating gives:

• \(2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3}\).

The reference angle \(\frac{\pi}{3}\) denotes an equivalent measure that resides within the first quadrant, making it easier to determine trigonometric function values.

cosine function

The cosine function is fundamental in trigonometry and relates to the x-coordinate of a point on the unit circle. For well-known reference angles, trigonometric values are readily accessible and often memorized. For instance, at \(\frac{\pi}{3}\), the cosine value is:

• \(\cos \frac{\pi}{3} = \frac{1}{2}\)

When recalling such values, it's important to consider the angle's location or quadrant in the unit circle. Elements of symmetry and reflection are crucial, as they determine the cosine's sign. In the fourth quadrant, where \(\frac{5\pi}{3}\) is located, the cosine function remains positive because the x-values are positive. Thus,

• \(\cos \frac{5\pi}{3} = \frac{1}{2}\)

Understanding these relationships enhances problem-solving skills involving trigonometric functions.

secant function

The secant function is the reciprocal of the cosine function. While the cosine of an angle relates to the x-coordinate on the unit circle, the secant gives a measure of how far this value is from the origin. For angles where cosine is positive,

• For example, \(\sec \theta = \frac{1}{\cos \theta}\)

For \(\frac{5\pi}{3}\), a positive cosine value \(\frac{1}{2}\) leads to a secant calculation:

• \(\sec \frac{5\pi}{3} = \frac{1}{\cos \frac{5\pi}{3}} = \frac{1}{\frac{1}{2}} = 2\)

Having reliable knowledge of trigonometric identities and their reciprocals, like secant, is invaluable for finding exact values in trigonometry.