Problem 234 - Precalculus

Question

For the following exercises, use a graphing calculator to evaluate. $$ \sec \frac{3 \pi}{4} $$

Step-by-Step Solution

Answer

\( \sec \frac{3 \pi}{4} = -\sqrt{2} \)

Step 1: Understand the Problem

We are asked to evaluate \( \sec \frac{3 \pi}{4} \). The secant function \( \sec(x) \) is the reciprocal of the cosine function, \( \sec(x) = \frac{1}{\cos(x)} \). So, we need to find \( \frac{1}{\cos \frac{3 \pi}{4}} \).

Step 2: Find the Reference Angle

The angle \( \frac{3 \pi}{4} \) is in the second quadrant where the cosine is negative. The reference angle for \( \frac{3 \pi}{4} \) in radians is \( \pi - \frac{3 \pi}{4} = \frac{\pi}{4} \).

Step 3: Calculate the Cosine of the Reference Angle

For the reference angle \( \frac{\pi}{4} \), \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). Since \( \frac{3 \pi}{4} \) is in the second quadrant, \( \cos \frac{3 \pi}{4} = -\frac{\sqrt{2}}{2} \).

Step 4: Evaluate the Secant Function

The secant is the reciprocal of the cosine, so \( \sec \frac{3 \pi}{4} = \frac{1}{\cos \frac{3 \pi}{4}} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} \).

Step 5: Simplify the Expression

Simplify \( -\frac{2}{\sqrt{2}} \) by multiplying the numerator and the denominator by \( \sqrt{2} \) to rationalize the denominator. This results in \( -\frac{2 \cdot \sqrt{2}}{2} = -\sqrt{2} \).

Key Concepts

Secant Function

The secant function is one of the six basic trigonometric functions, often abbreviated as "sec." It is defined as the reciprocal of the cosine function. This means that for a given angle, if you know the value of the cosine, you can find the secant by taking its reciprocal.

• If \( heta \) is the angle, then the relationship is \( ext{sec}( heta) = \frac{1}{ ext{cos}( heta)} \).
• The secant function is undefined for angles where the cosine value is zero because division by zero is not possible.
• The secant function has the same sign as the cosine function in each quadrant.

Understanding the secant function is crucial for solving trigonometric equations, especially those involving multiple trigonometric functions. In this context, evaluating \( \sec \frac{3 \pi}{4} \) allows us to practice finding the secant from the cosine using the reciprocal relationship.

Reference Angle

A reference angle is the acute angle that a given angle makes with the x-axis. It is always between 0 and \( \frac{\pi}{2} \) radians (0 and 90 degrees). Reference angles help simplify trigonometric calculations because they relate any angle in the unit circle to an equivalent acute angle, allowing us to apply known trigonometric values.

• The reference angle is particularly useful for finding trigonometric function values for angles outside the first quadrant.
• In our example, the angle \( \frac{3 \pi}{4} \) lies in the second quadrant. The reference angle is calculated as \( \pi - \frac{3 \pi}{4} = \frac{\pi}{4} \).
• With the reference angle, we can determine the cosine value easily and subsequently find the secant.

This concept helps to standardize the problem-solving approach in trigonometry and reduces complexity when working with angles in different quadrants.

Cosine Function

The cosine function is one of the primary trigonometric functions, essential for understanding relationships in right triangles and circles. It relates the adjacent side to the hypotenuse in a right triangle for a given angle. The cosine value ranges from -1 to 1, depending on the quadrant and angle.

• For an angle \( \theta \), \( \cos(\theta) \) gives the x-coordinate of the corresponding point on the unit circle.
• It is periodic with a period of \( 2\pi \), repeating its values every full rotation in radians.
• In the example with \( \frac{3 \pi}{4} \), the cosine of the reference angle \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \); however, \( \frac{3 \pi}{4} \) is in the second quadrant where the cosine is negative, making it \( -\frac{\sqrt{2}}{2} \).

The cosine function's properties are vital for transitioning from the triangle realm to the full scope of circular functions.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are those that are the "opposites" of the basic trigonometric functions. Specifically, secant, cosecant, and cotangent are the reciprocals of cosine, sine, and tangent, respectively.

• Each basic function has a reciprocal that is used when the original function appears in the denominator of a fraction in an equation.
• The reciprocal of the cosine function, the secant, is \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• Understanding these relationships helps in transforming equations and simplifying expressions.
• In this exercise, using the reciprocal of \( \cos \frac{3 \pi}{4} \) led to solving \( \sec \frac{3 \pi}{4} \) which simplifies to \( -\sqrt{2} \).

These reciprocal functions extend your toolbox for solving more complex trigonometric equations and understanding the connections among these mathematical concepts.