Problem 44 - Precalculus

Question

Use a calculator to evaluate the trigonometric functions for the indicated angle values. Round your answers to four decimal places. $$\sec \left(\frac{4 \pi}{9}\right)$$

Step-by-Step Solution

Answer

\( \sec\left(\frac{4\pi}{9}\right) \approx 5.7596 \).

Step 1: Convert to Cosine

Understand that the secant function is the reciprocal of the cosine function. Therefore, we need to find \( \cos\left(\frac{4\pi}{9}\right) \) first, and the secant will be the reciprocal of this cosine value.

Step 2: Calculate Cosine Value

Use a calculator to determine \( \cos\left(\frac{4 \pi}{9}\right) \). Ensure that your calculator is set to radian mode. Calculating this gives \( \cos\left(\frac{4\pi}{9}\right) \approx 0.1736 \).

Step 3: Calculate Secant Value

Since \( \sec(x) = \frac{1}{\cos(x)} \), take the reciprocal of the calculated cosine value. Therefore, \( \sec\left(\frac{4\pi}{9}\right) = \frac{1}{0.1736} \).

Step 4: Final Calculation

Perform the calculation \( \frac{1}{0.1736} \) to obtain \( \sec\left(\frac{4\pi}{9}\right) \approx 5.7596 \). Round your answer to four decimal places as required.

Key Concepts

Secant Function

The secant function, often abbreviated as \( \sec \), is one of the primary trigonometric functions and is defined as the reciprocal of the cosine function. This means that the secant of an angle \( x \) is equal to \( \frac{1}{\cos(x)} \). Simply put, instead of finding the straightforward cosine value of an angle, you would essentially "flip" the value to get the secant. For example, if the cosine of an angle is \( 0.5 \), the secant would be \( \frac{1}{0.5} = 2 \). This function is especially useful in cases where angles cannot be represented easily by sine or cosine alone. Some of the key properties include:

• It is undefined whenever the cosine value is zero, as division by zero is not possible.
• The secant function is positive when cosine is positive and negative when cosine is negative.
• It has a periodicity of \( 2\pi \), similar to cosine.

Secant functions often appear in calculus, particularly when solving integrals or differential equations.

Reciprocal of Cosine

Understanding the reciprocal of the cosine function is crucial to mastering the secant function. Since the secant is defined as the reciprocal, for any angle \( x \), \( \sec(x) = \frac{1}{\cos(x)} \).
This relationship shows the intrinsic connection between the two trigonometric functions. Whenever you have a cosine value, simply inverting this value, given it is not zero, will provide the secant result. Key points on this topic include:

• Reciprocal functions are based on this idea of inversion, which is a typical operation in mathematics.
• Any non-zero number divided into 1 provides a reciprocal outcome.
• This relation means that whenever the cosine value approaches zero, the secant value will increase significantly, often heading towards infinity.

Understanding the reciprocal helps in calculating secant values, especially when approaching boundaries where cosine approaches zero.

Radian Mode

When handling trigonometric functions such as the secant, it is vital to ensure your calculator is set to the correct mode. This is often radian mode, which treats angle measurements differently than degree mode. Specifically, radian is a unit based on the radius of a circle. Here are some essential aspects of radian mode:

• One complete circle is \( 2\pi \) radians, equivalent to 360 degrees.
• Common angles such as \( \pi \) and \( \frac{\pi}{2} \) equate to 180 and 90 degrees, respectively.
• Radian mode is frequently preferred in higher mathematics for its natural occurrence in integral and differential calculus.

To avoid calculation errors, especially in exercises like determining \( \sec\left(\frac{4\pi}{9}\right) \), remember to switch your calculator to radian mode before proceeding with any trigonometric evaluations.