Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact value. (a) \(\sec (2 \pi / 3)\) (b) \(\sec (-\pi / 6)\)

Step-by-Step Solution

Answer

(a) -2 (b) \(\frac{2\sqrt{3}}{3}\)

Step 1: Understand the Function

Secant is the reciprocal of the cosine function. To find \( \sec(\theta) \), we need the value of \( \cos(\theta) \), because \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

Step 2: Find \( \cos(2\pi/3) \)

The angle \( \frac{2\pi}{3} \) radians is located in the second quadrant of the unit circle, where cosine is negative. The reference angle is \( \pi/3 \). The value of \( \cos(\pi/3) \) is \( \frac{1}{2} \), so \( \cos(2\pi/3) = -\frac{1}{2} \).

Step 3: Calculate \( \sec(2\pi/3) \)

Use the formula \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Substitute \( \cos(2\pi/3) = -\frac{1}{2} \) into the equation: \( \sec(2\pi/3) = \frac{1}{-\frac{1}{2}} = -2 \).

Step 4: Find \( \cos(-\pi/6) \)

The angle \( -\frac{\pi}{6} \) radians is equivalent to \( \frac{11\pi}{6} \) radians (since \( -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6} \)). This angle is in the fourth quadrant, where cosine is positive. The reference angle is \( \frac{\pi}{6} \). The value of \( \cos(\pi/6) \) is \( \frac{\sqrt{3}}{2} \), so \( \cos(-\pi/6) = \frac{\sqrt{3}}{2} \).

Step 5: Calculate \( \sec(-\pi/6) \)

Using \( \sec(\theta) = \frac{1}{\cos(\theta)} \), substitute \( \cos(-\pi/6) = \frac{\sqrt{3}}{2} \) into the equation: \( \sec(-\pi/6) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \). To rationalize, multiply the numerator and denominator by \( \sqrt{3} \), resulting in \( \frac{2\sqrt{3}}{3} \).

Key Concepts

Secant in Trigonometry

Secant is one of the fundamental trigonometric functions you will encounter, often denoted as \( \sec(\theta) \). Understanding secant can be easy when you remember it is the reciprocal of the cosine function:
\[ \sec(\theta) = \frac{1}{\cos(\theta)} \]

• If \( \cos(\theta) = 0 \), \( \sec(\theta) \) is undefined because the reciprocal of zero does not exist.
• Secant can be either positive or negative depending on the quadrant in which the angle \( \theta \) lies.

Exploring the properties of secant, especially the way it changes across different quadrants, enhances your grasp of trigonometry. Remember, angles in radians can sometimes be trickier, so practicing with conversions is helpful!

Exploring the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It's a crucial tool in bridging angles and trigonometric functions. Each angle along the unit circle corresponds to a point with coordinates \( (\cos(\theta), \sin(\theta)) \).

• Angles in the unit circle are typically measured in radians, which divides the circle into sections based on \( \pi \).
• The circle is divided into four quadrants, and the sign of trigonometric functions can change depending on which quadrant the angle is in.

For instance, in the second quadrant, cosine values are negative, hence \( \sec(2\pi/3) \) ends up being negative because it's the reciprocal of a negative cosine.
Recognizing where an angle falls on the unit circle helps in determining the magnitude and sign of trigonometric functions like secant.

Understanding Reference Angles

Reference angles are another vital concept, providing a straightforward way of understanding larger or negative angles by relating them to angles in the first quadrant. The reference angle is the acute angle formed by dropping a perpendicular from the terminal side of the angle to the x-axis, essentially providing a 'reference' to help simplify the calculation of trigonometric functions.

• To find a reference angle \( \theta_{ref} \) for an angle \( \theta \):
  • If \( \theta \) is in the second quadrant: \( \theta_{ref} = \pi - \theta \).
  • If \( \theta \) is in the fourth quadrant: \( \theta_{ref} = 2\pi - \theta \).
• Reference angles can simplify finding trigonometric values because they reduce the problem to finding values involving familiar angles \( \leq \frac{\pi}{2} \).

For example, both \( \sec(2\pi/3) \) and \( \sec(-\pi/6) \) can be resolved by finding their corresponding reference angles which lead us to common values, allowing us to quickly compute things like cosine and secant.