Problem 15 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function at the given real number. (a) \(\sec \frac{11 \pi}{3}\) (b) \(\csc \frac{11 \pi}{3}\) (c) \(\sec \left(-\frac{\pi}{3}\right)\)

Step-by-Step Solution

Answer

(a) 2, (b) \( \frac{2\sqrt{3}}{3} \), (c) 2

Step 1: Simplify the Angle for (a)

To find \( \sec \frac{11\pi}{3} \), first simplify \( \frac{11\pi}{3} \) by subtracting multiples of \( 2\pi \) to find a coterminal angle. Calculate \( \frac{11\pi}{3} - 2\pi = \frac{11\pi}{3} - \frac{6\pi}{3} = \frac{5\pi}{3} \). This still exceeds \( 2\pi \), so subtract \( 2\pi \) again: \( \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3} \). Thus, \( \frac{11\pi}{3} \) is coterminal to \(-\frac{\pi}{3}\).

Step 2: Calculate Secant for (a)

Now find \( \sec \left( -\frac{\pi}{3} \right) \). Remember, \( \sec \theta = \frac{1}{\cos \theta} \). The cosine of \( -\frac{\pi}{3} \) is the same as the cosine of \( \frac{\pi}{3} \), which is \( \frac{1}{2} \). Therefore, \( \sec \left( -\frac{\pi}{3} \right) = \frac{1}{\frac{1}{2}} = 2 \).

Step 3: Simplify the Angle for (b)

To find \( \csc \frac{11\pi}{3} \), use the previously simplified angle, \( -\frac{\pi}{3} \), because \( \frac{11\pi}{3} \) is coterminal to it.

Step 4: Calculate Cosecant for (b)

To find \( \csc \left( -\frac{\pi}{3} \right) \), recall that \( \csc \theta = \frac{1}{\sin \theta} \). The sine of \( -\frac{\pi}{3} \) is the same as the sine of \( \frac{\pi}{3} \), which is \( \frac{\sqrt{3}}{2} \). Therefore, \( \csc \left( -\frac{\pi}{3} \right) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \). Simplify by rationalizing the denominator to get \( \frac{2\sqrt{3}}{3} \).

Step 5: Simplify the Angle for (c)

The angle for (c) is \( -\frac{\pi}{3} \). This angle needs no further simplification, as it is already between \( -\pi \) and \( \pi \).

Step 6: Calculate Secant for (c)

Use the calculation for \( \sec \left( -\frac{\pi}{3} \right) \) obtained in Step 2. Thus, \( \sec \left( -\frac{\pi}{3} \right) = 2 \).

Key Concepts

Secant Function

The secant function, often represented as \( \sec \theta \), is a fundamental trigonometric function that reciprocates the value of the cosine function. Its expression is formulated as:

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

This means that the secant of an angle is the reciprocal of the cosine of that angle. For instance, when dealing with angles in trigonometry, you might encounter an angle such as \( -\frac{\pi}{3} \). To find the secant of \( -\frac{\pi}{3} \), follow these steps:1. Identify the cosine value of the angle: \( \cos(-\frac{\pi}{3}) \) is the same as \( \cos(\frac{\pi}{3}) = \frac{1}{2} \).2. Use the reciprocal relationship: \( \sec(-\frac{\pi}{3}) = \frac{1}{\frac{1}{2}} = 2 \).When calculating the secant for angles more than \( 2\pi \), it is often necessary to first find the coterminal angle that lies within \( -2\pi \) and \( 2\pi \) as done with \( \sec \frac{11\pi}{3} \). Understanding and applying the secant function is crucial for solving trigonometric equations easily.

Cosecant Function

The cosecant function, symbolized by \( \csc \theta \), is another fundamental trigonometric function derived as the reciprocal of the sine function. To define it, we use:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Essentially, the cosecant of an angle is the inverse of its sine. Let us examine \( \csc(-\frac{\pi}{3}) \) for instance:1. Determine the sine value: \( \sin(-\frac{\pi}{3}) \) is equivalent to \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \).2. Find the reciprocal: \( \csc(-\frac{\pi}{3}) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2\sqrt{3}}{3} \), after rationalizing the denominator.When dealing with larger angles like \( \frac{11\pi}{3} \), you must first simplify the angle to within one cycle of \( 360^\circ \) or \( 2\pi \) by finding a coterminal angle, as illustrated in the exercise. The cosecant function provides vital utility in analyzing waveforms and solving trigonometric problems.

Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but differ by whole multiples of \( 2\pi \) (or \( 360^\circ \)). These angles can be located by either adding or subtracting \( 2\pi \) until the angle lies within a single trigonometric cycle, typically between \( 0 \) and \( 2\pi \) or \( -\pi \) and \( \pi \).For example, if you are given an angle like \( \frac{11\pi}{3} \), you can simplify it by subtracting \( 2\pi \) (or spreading it within \( 6\pi/3 \)) iteratively:- First step: \( \frac{11\pi}{3} - 2\pi = \frac{11\pi}{3} - \frac{6\pi}{3} = \frac{5\pi}{3} \).- Simplify further since \( \frac{5\pi}{3} \) exceeds \( 2\pi \): \( \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3} \).Thus, \( \frac{11\pi}{3} \) is coterminal to \(-\frac{\pi}{3}\). Recognizing coterminal angles simplifies the evaluation of trigonometric functions, permitting calculations within known reference angles and ensuring comprehension of periodicity in trigonometric contexts.