Problem 30 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \tan 270^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \tan 270^{\circ} \) is undefined.

Step 1: Recall the Tangent Function

The tangent function, \( \tan \theta \), is defined as the ratio of the opposite side to the adjacent side in a right triangle, or \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Its values are periodic with period \(360^{\circ}\).

Step 2: Determine the Cosine and Sine at 270°

At \(270^{\circ}\), which is on the negative y-axis, \( \sin 270^{\circ} = -1\) and \( \cos 270^{\circ} = 0\).

Step 3: Evaluate the Tangent at 270°

Using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we find \( \tan 270^{\circ} = \frac{-1}{0} \).

Step 4: Interpret the Result

Since division by zero is undefined, \( \tan 270^{\circ} \) does not have a finite value. Therefore, \( \tan 270^{\circ} \) is undefined.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the ratios of its sides. These functions are essential tools in both geometry and calculus. The three primary trigonometric functions are sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). Each function is associated with the angles and sides of a right triangle.

• The sine function (\( \sin \theta \)) is the ratio of the opposite side to the hypotenuse.
• The cosine function (\( \cos \theta \)) is the ratio of the adjacent side to the hypotenuse.
• The tangent function (\( \tan \theta \)) is defined as the ratio of the opposite side to the adjacent side. Alternately, it can also be expressed as the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

These functions are periodic, meaning they repeat their values in predictable cycles. For sine and cosine, this period is \(360^{\circ}\) while for tangent, the period is \(180^{\circ}\). Understanding these functions is vital for solving problems involving angles and lengths in triangles.

Undefined Values

An undefined value occurs in mathematics when a calculation cannot produce a meaningful result. In trigonometric functions, the most common scenario for encountering undefined values is during division by zero. The tangent function, specifically, is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
When the cosine of an angle is zero, the tangent becomes undefined because it involves division by zero. Consider the angle \(270^{\circ}\):

• Here, \(\cos 270^{\circ} = 0\).
• Since \(\tan 270^{\circ} = \frac{-1}{0}\), the tangent of \(270^{\circ}\) is not defined.

Understanding the conditions under which trigonometric functions become undefined helps in accurately working with angles and avoiding misconceptions.

Unit Circle

The unit circle is a fundamental concept in trigonometry and is typically used for defining the trigonometric functions for all angles. It is a circle with a radius of one centered at the origin \((0, 0)\) of a coordinate plane.

• Angles on the unit circle are measured from the positive x-axis, counter-clockwise.
• The coordinates of any point on this circle are used to represent the cosine and sine of the angle formed.
• This means for any angle \(\theta\), the coordinates \((\cos \theta, \sin \theta)\) give the cosine and sine values of that angle.

For \(270^{\circ}\), this angle is located on the negative y-axis, where the coordinates are \((0, -1)\). This helps us understand that \(\cos 270^{\circ} = 0\) and \(\sin 270^{\circ} = -1\). The unit circle is a crucial tool in understanding how trigonometric functions behave in different quadrants.

Sine and Cosine Ratios

Sine and cosine are two primary trigonometric functions that are often taught together due to their integral relationship.

• The sine of an angle \(\theta\) is the y-coordinate of its corresponding point on the unit circle, and
  li>The cosine is the x-coordinate.

These two values are used extensively in different trigonometric identities, one of which is crucial for evaluating the tangent function: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
For \(270^{\circ}\):

• \(\sin 270^{\circ} = -1\)
• \(\cos 270^{\circ} = 0\)

This direct relationship between sine and cosine through the unit circle is a core aspect, impacting other laws and theorems in trigonometry used to solve problems involving angles and lengths.