Problem 32 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \sec 100^{\circ} $$

Step-by-Step Solution

Answer

The value of \( \sec 100^{\circ} \) is approximately -5.7590.

Step 1: Understanding the Secant Function

The secant function, \( \sec \theta \), is the reciprocal of the cosine function. This means that \ \sec \theta = \frac{1}{ \cos \theta} \ where \ \theta \ is the angle in question.

Step 2: Convert the Angle to Radians

Mathematical computations such as trigonometric calculations are typically done in radians. To convert degrees to radians, use the formula: \[\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}\]For \ 100^{\circ} \the conversion is: \[100^{\circ} \times \frac{\pi}{180} = \frac{5\pi}{9} \] This is the angle in radians.

Step 3: Calculate the Cosine Value

Using a calculator, find the cosine of \ \frac{5\pi}{9} \ radians. \[\cos \left( \frac{5\pi}{9} \right) \approx -0.1736\] This is the approximate cosine value for \ 100^{\circ}. \

Step 4: Calculate the Secant Value

The secant value is the reciprocal of the cosine value. Therefore, take the reciprocal of the cosine value obtained:\[\sec \left(100^{\circ}\right) = \frac{1}{-0.1736} \approx -5.7590\]Thus, the secant of \ 100^{\circ} \ is approximately \ -5.7590 \ to four decimal places.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), plays an important role in trigonometry. It is defined as the reciprocal of the cosine function. So mathematically, you can express it as:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• "\( \theta \)" represents the angle you are interested in.

Understanding this relationship is key to solving problems involving the secant function. This simple reciprocal relationship allows you to convert between these functions easily. This is particularly useful when working with angles expressed in various units.

Radians Conversion

Trigonometric functions are typically calculated in radians rather than degrees. This means converting angles measured in degrees to radians is a crucial step in many mathematical computations.
To convert degrees to radians, use the formula:

• \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \)

For example, let's convert \(100^{\circ}\) to radians:

• \(100^{\circ} \times \frac{\pi}{180} = \frac{5\pi}{9} \)

This approach ensures that your angle is properly adjusted for trigonometric calculations performed in radians.

Cosine Value

Calculating the cosine value is another step in finding the secant of an angle. After converting your angle to radians, you can find the cosine using a calculator.
In the case of \(100^{\circ}\) converted to radians, which is \(\frac{5\pi}{9}\), the cosine value is approximately:

• \(\cos \left( \frac{5\pi}{9} \right) \approx -0.1736 \)

The cosine function returns a value based on the angle, which can vary between -1 and 1. Make sure to use a calculator set to radians for accurate results.

Reciprocal Function

The reciprocal function is vital in transforming between secant and cosine values. In the context of trigonometry:

• The reciprocal of a number \( x \) is \( \frac{1}{x} \).

When finding the secant of an angle, you use the cosine value. Then, you find its reciprocal:

• For example, \( \sec \left(100^{\circ}\right) = \frac{1}{\cos \left( \frac{5\pi}{9} \right)} \)
• Given that \(\cos \left( \frac{5\pi}{9} \right) \approx -0.1736\), the reciprocal value is \( \approx -5.7590 \).

This technique is used to deduce secant values from cosine, enabling you to handle a variety of trigonometric challenges.