Problem 16 - Precalculus: Functions and Graphs

Question

Find the exact value. (a) sec \(135^{\circ}\) (b) \(\sec \left(-210^{\circ}\right)\)

Step-by-Step Solution

Answer

(a) sec 135° = -√2, (b) sec -210° = -2√3/3

Step 1: Understand the Secant Function

The secant (sec) of an angle in a right triangle is the reciprocal of the cosine function. That means \( \sec(\theta) = \frac{1}{\cos(\theta)} \). We will use this knowledge to find the values of secant by first finding the corresponding cosine values.

Step 2: Find the Cosine of 135°

135° is in the second quadrant where cosine is negative. The reference angle for 135° is 45°. Therefore, \( \cos(135^{\circ}) = -\cos(45^{\circ}) = -\frac{\sqrt{2}}{2} \).

Step 3: Calculate Sec(135°)

Using the reciprocal relationship, \( \sec(135^{\circ}) = \frac{1}{\cos(135^{\circ})} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2} \).

Step 4: Find the Cosine of -210°

-210° can be converted to a positive angle by adding 360° which results in 150°. 150° is in the second quadrant where cosine is negative. The reference angle for 150° is 30°. Thus, \( \cos(150^{\circ}) = -\cos(30^{\circ}) = -\frac{\sqrt{3}}{2} \).

Step 5: Calculate Sec(-210°)

Utilizing the reciprocal relationship again, \( \sec(-210^{\circ}) = \frac{1}{\cos(150^{\circ})} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} \). Simplifying, we rationalize the denominator: \( -\frac{2\sqrt{3}}{3} \).

Key Concepts

Secant Function

The secant function, often abbreviated as sec, is a fundamental concept in trigonometry. It is defined as the reciprocal of the cosine function. Understanding this reciprocal relationship is key. With the secant defined as \( \sec(\theta) = \frac{1}{\cos(\theta)} \), it becomes clear that if you know the cosine of an angle, you can easily determine its secant by taking the reciprocal.

Let's explore this with an example: consider an angle \( \theta \). If \( \cos(\theta) = \frac{1}{2} \), then the secant of this angle would be \( \sec(\theta) = \frac{1}{1/2} = 2 \).

For students trying to grasp the secant function, it's often helpful to remember its connection to the unit circle, where the secant represents a line segment from the origin, perpendicular to the x-axis, and extending to the curve of the circle itself. This can help visualize why the secant is larger as the angle increases, especially when around 90° or 270° where cosine is zero, making secant undefined due to division by zero.

Cosine Function

The cosine function is another cornerstone of trigonometry, representing the x-coordinate of a point on the unit circle at a given angle. When people mention cosines, they often think of right triangles. In a right triangle, the cosine of an angle \( \theta \) is the ratio of the length of the adjacent side to the hypotenuse.

In unit circle terms, if you have an angle \( \theta \), the cosine of this angle is simply the x-coordinate of the point where the terminal side of the angle intersects the unit circle. The cosine function is periodic with a period of \( 360^{\circ} \) or \( 2\pi \) radians.

A crucial feature of cosine is its sign, which varies depending on the quadrant. For angles like 135°, since they lie in the second quadrant, the cosine would be negative. This can be seen from the step by step calculation for finding \( \cos(135^{\circ}) \), which uses the reference angle of 45° to establish \( \cos(135^{\circ}) = -\cos(45^{\circ}) = -\frac{\sqrt{2}}{2} \). Understanding these quadrant-based sign rules is very important in trigonometry.

Reciprocal Identities

Reciprocal identities are fundamental relationships in trigonometry that allow the expression of one trigonometric function in terms of another's reciprocal. These identities include:

• Secant as the reciprocal of cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• Cosecant as the reciprocal of sine: \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
• Cotangent as the reciprocal of tangent: \( \cot(\theta) = \frac{1}{\tan(\theta)} \)

Reciprocal identities simplify calculations and problem solving in trigonometry. For example, finding \( \sec(-210^{\circ}) \) in our exercise becomes straightforward once you calculate \( \cos(150^{\circ}) = -\frac{\sqrt{3}}{2} \). You apply the reciprocal identity to get \( \sec(-210^{\circ}) = \frac{1}{\cos(150^{\circ})} = -\frac{2}{\sqrt{3}} \).

Understanding and using these reciprocal identities is crucial, as they are often used to simplify equations and solve trigonometric problems. They also illustrate how interconnected the various trigonometric functions are.