Problem 20 - Precalculus

Question

Indicate the quadrant in which the terminal side of \(\theta\) must lie in order for the information to be true. \(\sec \theta\) and \(\csc \theta\) are both negative.

Step-by-Step Solution

Answer

The angle \( \theta \) lies in the third quadrant (Q3).

Step 1: Understanding Secant and Cosecant

The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, \( \sec \theta \) is negative when \( \cos \theta \) is negative. Similarly, the cosecant function, \( \csc \theta \), is the reciprocal of the sine function: \( \csc \theta = \frac{1}{\sin \theta} \). Thus, \( \csc \theta \) is negative when \( \sin \theta \) is negative.

Step 2: Determine Negative Cosine

The cosine of an angle is negative when the angle's terminal side is in either the second quadrant (Q2) or the third quadrant (Q3).

Step 3: Determine Negative Sine

The sine of an angle is negative when the angle's terminal side is in either the third quadrant (Q3) or the fourth quadrant (Q4).

Step 4: Find the Common Quadrant

To satisfy the conditions that both \( \sec \theta \) and \( \csc \theta \) are negative, we need the terminal side of \( \theta \) to be in a quadrant where both \( \cos \theta \) and \( \sin \theta \) are negative. From the previous steps, we can see that this only occurs in the third quadrant (Q3).

Key Concepts

Secant Function

The secant function, represented by \( \sec \theta \), is a key player in trigonometry. It's essentially the reciprocal of the cosine function. This means that \( \sec \theta = \frac{1}{\cos \theta} \). In simple terms, if you know the cosine of an angle, just flip it upside down to get the secant.
Secant values are negative when the cosine values are negative. This occurs in the second and third quadrants.
Why is that? Well, within the unit circle, the cosine corresponds to the x-coordinate, and x-coordinates are negative when the angle is in either of these quadrants.

• In the second quadrant, angles vary between 90° to 180°, and they return negative cosines.
• In the third quadrant, angles vary from 180° to 270°, also bringing negative cosine results.

Whenever you are asked to find when \( \sec \theta \) is negative, remember these quadrants!

Cosecant Function

The cosecant function, denoted by \( \csc \theta \), can be a bit tricky, but here's a simple breakdown. Like the secant function, the cosecant is a reciprocal, specifically of the sine function, so \( \csc \theta = \frac{1}{\sin \theta} \).
Pretty straightforward, right? If you have the sine of an angle, flip that value, and you get the cosecant. Now, the cosecant is negative when the sine itself is negative.
This happens in the third and fourth quadrants.

• In the third quadrant, you find angles between 180° to 270°, where the sine takes on negative values.
• In the fourth quadrant, from 270° to 360°, sine values remain negative as well.

Thus, for \( \csc \theta \) to be negative, look towards these two quadrants for solutions.

Unit Circle

The unit circle is a foundational tool in understanding trigonometry. Imagine a circle with its center at the origin (0,0) in the coordinate plane and a radius of 1.
This means every point on the circle satisfies the equation \(x^2 + y^2 = 1\).
But why is this circle so important? For one, it connects coordinates with angle measures in a very direct way.
Every angle \(\theta\) has a corresponding point \((x,y)\) on the circle. In trig terms:

• The x-coordinate represents \(\cos \theta\).
• The y-coordinate stands for \(\sin \theta\).

Each quadrant of the unit circle tells us something about what our trigonometric functions — sine, cosine, tangent, and their reciprocals — look like based on the signs of these coordinates.
For example, in the third quadrant, both the x and y values are negative. This means both \( \cos \theta \) and \( \sin \theta \) are negative there, making \( \sec \theta \) and \( \csc \theta \) negative as well. Hence, knowing the unit circle gives a clear path to understanding which quadrants meet specific trigonometric criteria.