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Chapter 1: Problem 22

Evaluate the following without a calculator. Some of these expressions are undefined. $$ \tan \left(270^{\circ}\right) $$

Short Answer

Expert verified
\( \tan(270^\text{掳}) \) is undefined because it involves division by zero.

Step by step solution

01

Understand the tangent function

The tangent of an angle \(\theta\) in a right-angled triangle is defined as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). However, it's more useful here to know tangent in terms of the unit circle: \( \tan(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)} \).
02

Convert angle to standard position

270 degrees is already in standard position (between 0 and 360 degrees), so no conversion is necessary.
03

Determine sin(270掳) and cos(270掳)

On the unit circle, \( \theta = 270^\text{掳} \) corresponds to the point \( (0, -1) \). This makes \( \text{sin}(270^\text{掳}) = -1 \) and \( \text{cos}(270^\text{掳}) = 0 \).
04

Compute tan(270掳)

Using the identity \( \tan(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)} \), substitute the values: \( \tan(270^\text{掳}) = \frac{\text{sin}(270^\text{掳})}{\text{cos}(270^\text{掳})} = \frac{-1}{0} \). Since division by zero is undefined, \(\tan(270^\text{掳})\) is undefined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Circle
To understand the tangent function, it's essential to know the unit circle. The unit circle is a circle with a radius of one, centered at the origin of a coordinate system.

Every point on this circle represents an angle, with its coordinates \( (x, y) \) corresponding to the values of \( cos(\theta) \) and \( sin(\theta) \), respectively.
  • The angle \( \theta \) is measured from the positive x-axis.
  • For example, at \( \theta = 270^{\circ} \), the coordinates on the unit circle are (0, -1).

This means:
  • \( cos(270^{\circ}) = 0 \)
  • \( sin(270^{\circ}) = -1 \)
The unit circle helps visualize trigonometric identities and understand the behavior of functions like tangent.
Trigonometric Identities
Trigonometric identities are fundamental equations in trigonometry that relate the angles and ratios of a triangle.
These can simplify complex trigonometric expressions and solve equations more easily.

For the tangent function, the key identity is:
\( \tan(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)} \)
This identity shows the relationship between the tangent function and the sine and cosine functions.
Applying this to the unit circle values:
  • \( \tan(270^{\circ}) = \frac{\text{sin}(270^{\circ})}{\text{cos}(270^{\circ})} \)
  • Substituting the values: \( \tan(270^{\circ}) = \frac{-1}{0} \)
This equation shows that tangent is the ratio of sine to cosine and its value can be undefined if cosine is zero.
Undefined Expressions
Understanding undefined expressions is crucial in trigonometry because certain operations cannot be performed on specific values.

In the context of tangent, an undefined expression occurs when we try to divide by zero.
This is because division by zero is mathematically undefined and does not produce a finite result.

For example, at \( \theta = 270^{\circ} \), on the unit circle:
  • \

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