Problem 7 - Algebra 2

Question

Find the exact value of each trigonometric function. \(\sec \frac{7 \pi}{6}\)

Step-by-Step Solution

Answer

\(\sec \frac{7\pi}{6} = -\frac{2\sqrt{3}}{3}\)

Step 1: Convert Radians to Degrees

First, let's convert the angle from radians to degrees. The angle given is \(\frac{7\pi}{6}\). We can use the conversion factor \(180^\circ = \pi\, \text{radians}\) to convert:\[\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ\] So, \(\frac{7\pi}{6}\) radians is equal to \(210^\circ\).

Step 2: Identify the Quadrant

Now that we have the angle as \(210^\circ\), determine which quadrant it is in. An angle of \(210^\circ\) is in the third quadrant.

Step 3: Find Reference Angle

The reference angle is the angle's acute component with respect to the x-axis. For any angle \(\theta\) in the third quadrant, the reference angle is \(\theta - 180^\circ\). Thus:\[210^\circ - 180^\circ = 30^\circ\] Hence, the reference angle is \(30^\circ\).

Step 4: Determine the Secant Value

Secant is the reciprocal of cosine. The cosine of \(30^\circ\) is \(\frac{\sqrt{3}}{2}\). However, since \(210^\circ\) is in the third quadrant, cosine values are negative. Thus:\[\cos(210^\circ) = -\frac{\sqrt{3}}{2}\] Therefore, the secant, being the reciprocal of cosine, is:\[\sec(210^\circ) = -\frac{2}{\sqrt{3}}\] To rationalize the denominator:\[\sec(210^\circ) = -\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\]

Step 5: Finalize Solution

Now we have the exact value for the secant function:\[\sec \frac{7\pi}{6} = -\frac{2\sqrt{3}}{3}\] This is the exact value of \(\sec \frac{7\pi}{6}\).

Key Concepts

Radians to Degrees Conversion

When dealing with trigonometric functions, it is common to switch between radians and degrees. Radians and degrees are two units that measure angles, and knowing how to convert one to the other is essential. The conversion factor simplifies this process. Since a full circle equals 360 degrees or \(2\pi\) radians, we have the conversion factor \(180^\circ = \pi\) radians.

To convert radians to degrees, multiply the radian measure by \(\frac{180^\circ}{\pi}\). For example, converting \(\frac{7\pi}{6}\) radians to degrees, we perform the calculation:

\[\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ\]

This result shows us that \(\frac{7\pi}{6}\) radians is equivalent to 210 degrees, which is useful in identifying the angle's properties, such as its reference angle and quadrant.

Reference Angle

The reference angle is a helpful concept for working with angles in different quadrants. It measures the angle's smallest deviation from the closest x-axis, ensuring it is always between 0 and 90 degrees.

To find the reference angle for an angle \(\theta\) in degrees, consider its quadrant:

• For the first quadrant, the reference angle is \(\theta\).
• For the second quadrant, subtract \(\theta\) from 180 degrees.
• For the third quadrant, subtract 180 degrees from \(\theta\).
• For the fourth quadrant, subtract \(\theta\) from 360 degrees.

In our example, the angle 210 degrees lies in the third quadrant. Calculating the reference angle involves:

\[210^\circ - 180^\circ = 30^\circ\]

This means the reference angle is 30 degrees. Understanding this allows us to use known trigonometric values easily.

Secant Function

The secant function, represented as \(\sec \theta\), is one of the six primary trigonometric functions. It is the reciprocal of the cosine function, meaning \(\sec \theta = \frac{1}{\cos \theta}\). Understanding the secant function is crucial for various applications in mathematics and physics.

Secant can face undefined situations when the cosine is zero, as division by zero is undefined. In general, secant is positive in the first and fourth quadrants, and negative in the second and third quadrants. This comes from the sign of the cosine in those quadrants.

For the example angle 210 degrees:1. First, find \(\cos(210^\circ)\). Since cosine is negative in the third quadrant and \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), we have that:
\[\cos(210^\circ) = -\frac{\sqrt{3}}{2}\]2. Then, compute \(\sec(210^\circ)\) as its reciprocal:
\[-\frac{2}{\sqrt{3}}\]3. Finally, rationalize the denominator for a neater expression:
\[\sec(210^\circ) = -\frac{2\sqrt{3}}{3}\]

Reciprocal of Cosine

Understanding that secant is the reciprocal of cosine provides insight into its relationship with other trigonometric functions. This relationship is handy because the cosine function has various known values, especially in standard angles.

Given \(\sec \theta = \frac{1}{\cos \theta}\), if we know \(\cos \theta\), we can easily determine \(\sec \theta\) by reciprocating the value. This is particularly useful when dealing with common angles like 30°, 45°, and 60°, which have well-memorized cosine values.

• For 30°, \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), so \(\sec(30^\circ) = \frac{2}{\sqrt{3}}\).
• To rationalize, multiply the expression by \(\frac{\sqrt{3}}{\sqrt{3}}\), leading to \(\sec(30^\circ) = \frac{2\sqrt{3}}{3}\).

In practical applications, if cosine is negative (as in quadrants two and three), secant will also be negative due to its reciprocal nature, affecting calculations in physics and engineering contexts.