Problem 22 - Algebra 2 and Trigonometry

Question

If \(\sec \theta\) is undefined, find all possible values of \(\sin \theta\)

Step-by-Step Solution

Answer

The possible values of \(\sin \theta\) are 1 and -1.

Step 1: Understand When \(\sec \theta\) is Undefined

The secant function, \(\sec \theta\), is undefined wherever \(\cos \theta = 0\), since \(\sec \theta = \frac{1}{\cos \theta}\). This occurs when \(\theta = \frac{\pi}{2} + k\pi\) where \(k\) is any integer. These angles correspond to vertical lines on the unit circle where the cosine value is zero.

Step 2: Determine Sine Values at the Undefined Points

When \(\cos \theta = 0\), \(\theta = \frac{\pi}{2} + k\pi\). At these points on the unit circle, the sine function, \(\sin \theta\), has values of either 1 or -1, since they correspond to the top and bottom points of the unit circle, respectively.

Step 3: Summarize the Possible Sine Values

Given that \(\theta = \frac{\pi}{2} + k\pi\), the possible values for \(\sin \theta\) are 1 (when \(\theta = \frac{\pi}{2} + 2k\pi\)) and -1 (when \(\theta = \frac{3\pi}{2} + 2k\pi\)).

Key Concepts

Secant Function

The secant function, denoted as \(\sec \theta\), is one of the six primary trigonometric functions. It is defined as the reciprocal of the cosine function: \(\sec \theta = \frac{1}{\cos \theta}\). This means that \(\sec \theta\) can only be evaluated when \(\cos \theta\) is not zero, because dividing by zero is undefined.
This condition makes \(\sec \theta\) undefined wherever the cosine values hit zero. Understanding these points is crucial because it impacts how we interpret secant's behavior on the unit circle. Knowing when \(\sec \theta\) is undefined is the first step to understanding the problem.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 centered at the origin \((0,0)\) in a coordinate plane. The unit circle is vital for understanding trigonometric functions like sine, cosine, and secant. This is because the angles and their trigonometric values can be easily represented on this circle.
On the unit circle, every angle \(\theta\) corresponds to a point \((x, y)\) that represents \((\cos \theta, \sin \theta)\). For secant, we are particularly interested in vertical lines where \(\cos \theta = 0\). These lines occur at \(\theta = \frac{\pi}{2} + k\pi\), where \(k\) is any integer. It's at these points that secant becomes undefined.

Sine Values

On the unit circle, the sine of an angle \(\theta\) corresponds to the \(y\)-coordinate of the point where the angle intersects the circle. When \(\sec \theta\) is undefined, \(\cos \theta\) equals zero.
The angles that satisfy this are \(\theta = \frac{\pi}{2} + k\pi\), leading to points at the top and bottom of the unit circle. At these angles, \(\sin \theta\) can only be 1 or -1. More specifically, \(\sin \theta = 1\) when the angle corresponds to the top point \((0,1)\), and \(\sin \theta = -1\) when at the bottom \((0, -1)\). Understanding this relationship helps solve exercises like the one presented.

Cosine and Secant Relationship

The relationship between cosine and secant is a reciprocal one: \(\sec \theta = \frac{1}{\cos \theta}\). This means that whenever \(\cos \theta = 0\), \(\sec \theta\) becomes undefined. To find where the secant function is undefined, you need to find where the cosine function is zero.
These are the points along the unit circle where the angle \(\theta\) creates a vertical line, specifically at \(\frac{\pi}{2} + k\pi\) for any integer \(k\). Consequently, understanding where cosine becomes zero is essential to unveiling at which angles secant will fail to provide a value. Appreciating this connection helps in understanding broader trigonometric concepts and in solving relevant problems more efficiently.