Problem 16 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function. $$\text { sec } 300^{\circ}$$

Step-by-Step Solution

Answer

\(\sec 300^{\circ} = 2\)

Step 1: Determine the Reference Angle

The angle given is \(300^{\circ}\). Since secant is periodic with \(360^{\circ}\), we can find the reference angle by subtracting \(300^{\circ}\) from \(360^{\circ}\), which gives us a reference angle of \(60^{\circ}\).

Step 2: Identify the Quadrant

The angle \(300^{\circ}\) is in the fourth quadrant because it is between \(270^{\circ}\) and \(360^{\circ}\). In the fourth quadrant, the secant function (being the reciprocal of the cosine function) is positive.

Step 3: Calculate Secant of the Reference Angle

The secant function is the reciprocal of the cosine function. Since our reference angle is \(60^{\circ}\), we need to find \(\sec 60^{\circ}\). We know that \(\cos 60^{\circ} = \frac{1}{2}\), so \(\sec 60^{\circ} = \frac{1}{\cos 60^{\circ}} = 2\).

Step 4: Determine the Sign and Value of Sec 300°

Since we are in the fourth quadrant where secant is positive, the value of \(\sec 300^{\circ}\) is the same as \(\sec 60^{\circ}\). Therefore, \(\sec 300^{\circ} = 2\).

Key Concepts

Secant Function

The secant function, denoted as \( \sec \), is one of the fundamental trigonometric functions. It is the reciprocal of the cosine function. In mathematical terms, if \( \cos \theta = x \), then \( \sec \theta = \frac{1}{x} \). This means that wherever the cosine value is zero, the secant function is undefined, as division by zero is not possible.

The secant function is important in trigonometry for several reasons. One key aspect is its periodic nature, repeating every \( 360^{\circ} \) or \( 2\pi \) radians. This periodicity implies that \( \sec(\theta) = \sec(\theta + 360^{\circ}) \), allowing calculations to be simplified for large angles.

When evaluating the secant of an angle, always consider whether the angle must be reduced to its reference angle first. We'll elaborate on reference angles in the subsequent section.

Reference Angle

To find a trigonometric value like secant easily, often we reduce the given angle to what's called a "reference angle". The reference angle is fundamental for simplifying problems, as it represents the smallest positive angle that shares the terminal side with the original angle.

To find the reference angle, we generally:

• Subtract from \( 360^{\circ} \) if the angle is in the fourth quadrant.
• Subtract from \( 180^{\circ} \) if the angle is in the second or third quadrants.
• Use the angle itself if it lies in the first quadrant.

In the context of \( \sec 300^{\circ} \), the angle \( 300^{\circ} \) lies in the fourth quadrant. Thus, the reference angle is \( 360^{\circ} - 300^{\circ} = 60^{\circ} \). Calculating trigonometric functions using reference angles can simplify our process immensely.

Quadrant Analysis

Quadrant analysis is critical when determining the positivity or negativity of trigonometric function values. The coordinate plane is divided into four quadrants, each affecting the sign of trigonometric functions.

Here's how to determine signs based on quadrants:

• First Quadrant \((0^{\circ} \text{ to } 90^{\circ}):\) All trigonometric functions are positive.
• Second Quadrant \((90^{\circ} \text{ to } 180^{\circ}):\) Sine is positive; cosine and secant are negative.
• Third Quadrant \((180^{\circ} \text{ to } 270^{\circ}):\) Tangent is positive, sine and cosine (and their reciprocals) are negative.
• Fourth Quadrant \((270^{\circ} \text{ to } 360^{\circ}):\) Cosine and secant are positive; sine and tangent are negative.

For \( \theta = 300^{\circ} \), since it's in the fourth quadrant, the secant function \( \sec \theta \) is positive. Thus, knowing the quadrant helps in establishing the final sign of the trigonometric value.