Problem 16 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\tan (\pi/2)\) is undefined.

Step 1: Understand quadrantal angles

Quadrantal angles are angles in standard position with their terminal sides on the x-axis or y-axis. The four quadrantal angles are 0 (or 2\(\pi\)), \(\pi/2\), \(\pi\), and \(3\pi/2\).

Step 2: Understand trigonometric function at quadrantal angles

The three primary trigonometric functions sine, cosine, and tangent behave in a specific way at quadrantal angles. The tangent function, \( \tan x \), is equivalent to \( \sin x / \cos x \). So we need to find out the values of sine and cosine at \(\pi/2\).

Step 3: Evaluate sin and cos at \(\pi/2\)

At \(\pi/2\), the sine function equals 1 and the cosine function equals 0. That is \(\sin(\pi/2) = 1\) and \(\cos(\pi/2) = 0\).

Step 4: Evaluate tan at \(\pi/2\)

The function \(\tan x\) evaluates to \(\sin x / \cos x\). So, \(\tan (\pi/2) = 1/0\). But, in mathematics, division by zero is undefined.

Step 5: Conclusion

Therefore, \(\tan (\pi/2)\) is undefined.

Key Concepts

Quadrantal Angles

Quadrantal angles are very special in trigonometry. They occur when an angle's terminal side is aligned with either the x-axis or the y-axis. These include 0 (or 2\(\pi\)), \(\pi/2\), \(\pi\), and \(3\pi/2\). Each of these angles corresponds to a 90-degree rotation from the previous one.
At quadrantal angles, the trigonometric functions take on unique values, simplifying some calculations and complicating others. Understanding these angles is crucial for computing trigonometric functions correctly.

Tangent Function

The tangent function, represented as \(\tan x\), is one of the main functions in trigonometry. It can be visualized as the ratio of sine to cosine:
\[ \tan x = \frac{\sin x}{\cos x} \]
This relationship shows why the tangent function can behave strangely at quadrantal angles. Specifically, when the cosine of an angle is zero (as it is at \(\pi/2\)), the tangent function is undefined because division by zero is not possible.
This makes it important to comprehend how tangent operates, particularly with these angles.

Sine and Cosine Functions

Sine and cosine are foundational trigonometric functions that represent the coordinates of a point on the unit circle. For an angle \(x\):

• \(\sin x\) is the y-coordinate
• \(\cos x\) is the x-coordinate

At the quadrantal angle \(\pi/2\), these values are:

• \(\sin(\pi/2) = 1\)
• \(\cos(\pi/2) = 0\)

Recognizing these values helps to understand why \(\tan(\pi/2)\) becomes undefined, since you're dividing 1 by 0 in that expression.

Undefined Expression

In mathematics, certain expressions, like division by zero, are considered undefined. They cannot be rationally solved or simplified. When evaluating \(\tan(\pi/2)\), the expression becomes \(1/0\), which is not a valid operation.
This undefined nature ties back to how tangent relies on both sine and cosine. If the cosine is zero, division leads to an undefined outcome.
Understanding this concept is essential for dealing with expressions involving trigonometric functions at quadrantal angles.