Problem 53 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the giren trigonometric functions to four decimal places. $$\sec (3.2)$$

Step-by-Step Solution

Answer

The value of \( \sec(3.2) \) to four decimal places is -1.4896

Step 1: Understand the Secant Function.

The secant of an angle \( x \) in a right triangle is defined as the hypotenuse divided by the adjacent. This operation is reversed for the cosine function, which means that the secant is the reciprocal of the cosine.

Step 2: Using the Calculator.

Now that the task is understood, a scientific calculator should be used to calculate \( \sec (3.2) \).

Step 3: Doing the Operation

As secant is the reciprocal of cosine, calculate the cosine of 3.2 first and then take the reciprocal. The calculator should have this as a built-in function. Calculate the cosine of 3.2 in radian mode on the calculator and then reciprocate that value (1 divided by cosine of 3.2) to find the secant. Always remember to round to four decimal places.

Key Concepts

Secant Function

The secant function, denoted as \( \sec(x) \), is an important trigonometric function linked closely to the cosine function. Think of it like flipping the cosine function. It represents the ratio of the hypotenuse over the adjacent side in a right triangle. But in trigonometry, it goes beyond triangles using functions based on angles.

• Define \( \sec(x) \) as the reciprocal of the cosine function: \( \sec(x) = \frac{1}{\cos(x)} \).
• In a triangle, this is more naturally thought of in terms of its sides, but in functions, it can be calculated even when there isn’t a physical triangle.

Instead of worrying about measuring sides, just focus on the reciprocal aspect. This makes understanding complex angles or circles simpler.

Cosine Function

The cosine function, represented as \( \cos(x) \), is one of the fundamental functions in trigonometry. It reflects how the horizontal side of a right triangle behaves compared to the hypotenuse. But more generally, cosine tells us about the x-coordinate of a point on the unit circle.

• Cosine of an angle gives the ratio of the adjacent side over the hypotenuse.
• The unit circle helps visualize the cosine value as the horizontal distance from the origin to the point on the circle.

Thus, knowing the value of a cosine is helpful when dealing with the secant function. Since secant is the reciprocal, first find \( \cos(3.2) \), then to find \( \sec(3.2) \), use \( \sec(x) = \frac{1}{\cos(x)} \). This converts the cosine to secant with ease.

Scientific Calculator Usage

Using a scientific calculator properly can greatly simplify your work with trigonometric functions. These calculators can compute sine, cosine, tangent, and their reciprocals directly.

• Make sure your calculator is in the right mode—radians or degrees—depending on your angle measure. For \( \sec(3.2) \), the calculator should be in radian mode.
• To find \( \sec(3.2) \), first compute \( \cos(3.2) \) using the built-in function.
• Remember, process the reciprocal by pressing the appropriate button or manually computing \( \frac{1}{\cos(3.2)} \).

Ensure rounding is done to four decimal places for precision. Familiarizing yourself with these steps will improve your trigonometry skills and calculator efficiency.