Problem 13 - Algebra and Trigonometry

Question

evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$ \tan \frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

The value of \(\tan \frac{3 \pi}{2}\) is undefined because it implies division by zero.

Step 1: Identify the coordinates

The first step is to identify the coordinates on the unit circle that correspond to the given angle. The angle given is \(\frac{3 \pi}{2}\) radians, which is equivalent to 270 degrees. This corresponds to the point (0, -1) on the unit circle.

Step 2: Evaluate the tan function

Now that the coordinates on the unit circle have been identified, the tangent can be evaluated. Tangent is defined as the ratio of the y-coordinate to the x-coordinate. In other words, \(\tan \theta = \frac{y}{x}\). Plugging in the coordinates (0, -1) into the definition, we get \(\tan \frac{3 \pi}{2}= \frac{-1}{0}\).

Step 3: Evaluate the ratio

The last step is to evaluate the ratio \(\frac{-1}{0}\). When we have zero in the denominator of a fraction, it implies division by zero, which is an undefined mathematical operation.

Key Concepts

Quadrantal Angles

Quadrantal angles are special angles that lie along the main axes of the coordinate plane. These angles are typically measured from the positive x-axis in standard position and are integral multiples of 90 degrees, or \( \frac{\pi}{2} \) radians in trigonometric terms. There are four primary quadrantal angles:

• 0 degrees (or 0 radians)
• 90 degrees (or \( \frac{\pi}{2} \) radians)
• 180 degrees (or \( \pi \) radians)
• 270 degrees (or \( \frac{3\pi}{2} \) radians)

When solving trigonometric functions such as tangent at these angles, understanding their position on the coordinate plane is essential. These angles often result in special values or undefined operations because they coincide with the axes, where typically, either the x-coordinate or y-coordinate is zero. As seen in the exercise, \( \tan \frac{3 \pi}{2} \) is undefined due to division by zero, highlighting the importance of recognizing quadrantals and their coordinate characteristics.

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle with a radius of 1 centered at the origin of the coordinate plane. The key feature of the unit circle is that it provides a simple and consistent way to define trigonometric functions for any angle derived from the circle’s geometry.

Every point on the unit circle is described by coordinates \((x, y)\), which correspond to \( \cos(\theta) \) and \( \sin(\theta) \), respectively. Therefore, the coordinates for specific angles give direct values for these trigonometric functions:

• At 0 degrees (0 radians), the point is (1, 0)
• At 90 degrees (\( \frac{\pi}{2} \) radians), the point is (0, 1)
• At 180 degrees (\( \pi \) radians), the point is (-1, 0)
• At 270 degrees (\( \frac{3\pi}{2} \) radians), the point is (0, -1)

In the context of the unit circle, these points arise at quadrantal angles where tangent is particularly interesting as it involves \( \frac{y}{x} \). Understanding this geometric framework aids significantly in comprehending how trigonometric values are derived and computed.

Tangent Function

The tangent function is an important trigonometric function often defined as the ratio of the sine of an angle to the cosine of that angle. On the unit circle, this translates to the ratio of the y-coordinate to the x-coordinate. Mathematically, this is expressed as:

• \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x} \]

The tangent function is particularly dynamic, as it can take on values ranging from negative infinity to positive infinity, but it is undefined where the cosine equals zero (leading to division by zero).

For quadrantal angles such as \( 90^\circ \) and \( 270^\circ \) (or \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) radians), the tangent function encounters this undefined scenario because the x-coordinate is zero. Recognizing the conditions under which tangent becomes undefined is vital for solving trigonometric problems efficiently.

Thus, when evaluating the tangent function, it’s crucial to inspect the angle's position on the unit circle to avoid or acknowledge any "undefined" results due to division by zero.