Question

True or False \(\tan \frac{\pi}{2}\) is not defined.

Short answer

True

Step-by-step solution

Understand the Tangent Function

The tangent function is defined as the ratio of the sine and cosine functions. Mathematically, \(\tan \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\).

Determine \( \text{cos} \frac{\text{π}}{2} \)

Evaluate the cosine of \( \frac{\text{π}}{2} \). In the unit circle, \( \text{cos} \frac{\text{π}}{2} = 0 \).

Check for Division by Zero

Substitute \( \frac{\text{π}}{2} \) into the tangent function: \( \tan \frac{\text{π}}{2} = \frac{\text{sin} \frac{\text{π}}{2}}{\text{cos} \frac{\text{π}}{2}} = \frac{1}{0} \). Since division by zero is undefined, \( \tan \frac{\text{π}}{2} \) is not defined.

Confirm the Statement

Since we have determined that division by zero makes \( \tan \frac{\text{π}}{2} \) undefined, the statement ' \( \tan \frac{\text{π}}{2} \) is not defined.' is true.

Key concepts

headline of the respective core concept

The tangent function is one of the basic trigonometric functions. It's defined as the ratio of the sine function to the cosine function.
In simpler terms, for any given angle \(\theta\), we express the tangent function as:

\( \tan \theta = \frac{\text{sin} \theta}{\text{cos} \theta} \).
This means we take the value of the sine at that angle and divide it by the value of the cosine at the same angle. This function plays a crucial role in trigonometry and is often used to solve problems in various fields, such as physics, engineering, and navigation.
It's important to remember that the values of sine and cosine can change depending on the angle, which in turn changes the value of the tangent function. As the tangent function involves division, it can sometimes be undefined if the cosine of the angle is zero.

headline of the respective core concept

Trigonometric functions, including sine, cosine, and tangent, are fundamental to understanding angles and circles. These functions relate the angles of a triangle to the lengths of its sides. They also appear in various formulas and equations across different areas of math and science. The unit circle is a helpful concept for understanding these functions. It’s a circle with a radius of one centered at the origin of a coordinate plane.

Here are the primary trigonometric functions:

• Sine (sin): It measures the y-coordinate of the point on the unit circle at a given angle.


• Cosine (cos): It measures the x-coordinate of the point on the unit circle at a given angle.


• Tangent (tan): It is the ratio of sine to cosine, \( \tan \theta = \frac{\text{sin} \theta}{\text{cos} \theta} \).

Understanding these functions and their relationships makes it easier to solve triangles, model periodic phenomena, and comprehend waves and oscillations.

headline of the respective core concept

For some angles, trigonometric functions can be undefined due to division by zero. The tangent function specifically can encounter undefined values. This happens when the cosine of the angle equals zero because dividing by zero is undefined.

Let’s consider the specific example in the exercise:

• We need to evaluate \(\tan \frac{\pi}{2}\): According to the definition, \(\tan \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\).


• At \(\frac{\pi}{2}\), sin(\(\frac{\pi}{2}\)) = 1 and cos(\(\frac{\pi}{2}\)) = 0.


• So, \(\tan \frac{\pi}{2} = \frac{1}{0}\). Since dividing by zero is undefined, \(\tan \frac{\pi}{2}\) is not defined.

Understanding this concept is crucial not just for math classes but also in practical applications where undefined values must be handled carefully to avoid errors and inaccuracies.