Problem 239 - Precalculus

Question

For the following exercises, use a graphing calculator to evaluate. $$ \sec 310^{\circ} $$

Step-by-Step Solution

Answer

\( \sec 310^{\circ} \approx 1.305 \)

Step 1: Understanding Secant

The secant function, noted as \( \sec \theta \), is the reciprocal of the cosine function. This means \( \sec \theta = \frac{1}{\cos \theta} \). So, to find \( \sec 310^{\circ} \), we first need to find \( \cos 310^{\circ} \).

Step 2: Convert Degrees to Radians

Many graphing calculators require angles to be in radians for trigonometric functions. To convert degrees to radians, use the formula: \[ 310^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{31\pi}{18} \text{ radians} \]

Step 3: Calculate Cosine

Using a graphing calculator, find \( \cos(310^{\circ}) \) or \( \cos\left(\frac{31\pi}{18}\right) \). Ensure the calculator is in the correct mode (degrees or radians). For 310°, \( \cos 310^{\circ} \approx 0.7660 \).

Step 4: Find the Secant

Now calculate \( \sec 310^{\circ} \), which is the reciprocal of \( \cos 310^{\circ} \). Therefore: \[ \sec 310^{\circ} = \frac{1}{0.7660} \approx 1.305 \]

Key Concepts

Secant Function

The secant function is a fundamental concept in trigonometry. It's denoted as \( \sec \theta \) and is defined as the reciprocal of the cosine function.
In simpler terms, if you know the cosine of an angle, you can find the secant by taking the reciprocal (or inverse) of that value.
This means \( \sec \theta = \frac{1}{\cos \theta} \).

• The secant function is useful for solving problems where cosine values are known but a different perspective is needed.
• It often appears in geometry and calculus, especially when dealing with trigonometric identities.

One key thing to remember is that secant can be undefined when the cosine of an angle equals zero, as division by zero is not possible. This occurs for angles like 90° and 270° (or \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) radians), where the cosine is zero on the unit circle. Understanding these concepts will help you handle trigonometric problems effectively.

Cosine Function

At the heart of finding secant is knowing the cosine function. Cosine, abbreviated as \( \cos \theta \), tells us about the horizontal component of an angle in the unit circle.
This function takes an angle and returns a value ranging between -1 and 1, depending on where the angle lies on the circle.

• For angles in the standard position, the cosine value indicates how far left or right the endpoint of the angle is from the center of the circle.
• Cosine is crucial in determining the secant, since \( \sec \theta = \frac{1}{\cos \theta} \).

To find \( \cos 310^{\circ} \), for example, you could use either the cosine function directly or a graphing calculator. It’s always important to ensure your calculator is in the correct mode—degrees for this example—as cosine can yield different results in radians or degrees. If calculated correctly, \( \cos 310^{\circ} \) is approximately 0.7660.

Angle Conversion

Understanding how to convert between degrees and radians is a helpful skill in trigonometry. Some calculators and functions prefer angles in radians over degrees.
The conversion is straightforward using the formula: \[ \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180^{\circ}} \right). \]For 310°, converting it to radians would give you \( \frac{31\pi}{18} \). This ensures you're using the correct unit for your trigonometric calculations.

• Recognizing when and how to convert angles helps avoid calculation errors.
• Most trigonometric functions are periodic, meaning they repeat their values in cycles, which you can express more easily in radians.

By mastering degree and radian conversions, you'll be better prepared to approach problems across different contexts, both in mathematics and applied sciences.