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Chapter 7: Problem 7

Find an equivalent expression for each of the following. $$\tan \left(x-\frac{\pi}{2}\right)$$

Short Answer

Expert verified
The equivalent expression is \(-\text{cot}(x)\).

Step by step solution

01

Recall the Tangent Angle Subtraction Formula

The tangent of a difference of angles can be expressed using the formula: \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]
02

Set Up the Formula with Given Angles

Let \(a = x\) and \(b = \frac{\frac{\text{蟺}}{2}}{2}\). Substitute these into the tangent subtraction formula: \[ \tan \bigg(x - \frac{\text{蟺}}{2}\bigg) = \frac{\tan x - \tan \frac{\text{蟺}}{2}}{1 + \tan x \tan \frac{\text{蟺}}{2}} \]
03

Simplify Using Known Tangent Values

Recall that \( \tan \frac{\text{蟺}}{2} \to \text{undefined}\). Since this value makes the denominator zero, an alternative simpler trigonometric identity can be used: \[ \tan\bigg(x - \frac{\text{蟺}}{2}\bigg) = -\frac{1}{\tan x} \]
04

Rewrite the Expression

The expression can be rewritten as the cotangent function: \[ \tan\bigg(x - \frac{\text{蟺}}{2}\bigg) = -\frac{1}{\tan x} = -\text{cot} \big(x\big) \]

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value within their domains. These identities are useful for simplifying expressions and solving equations involving trigonometric functions.
Here are a few popular identities to remember:
  • Pythagorean identities: \(\text{sin}^2 x + \text{cos}^2 x = 1\)
  • Reciprocal identities: \(\text{cot} x = \frac{1}{\tan x}\)
  • Angle sum and difference identities, like \(\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\)
These identities can simplify complex trigonometric expressions and solve equations involving angles.
Angle Subtraction
Angle subtraction is one of the core concepts in trigonometry. When working with angles, you often need to subtract or add them. The tangent subtraction formula helps us find the tangent of the difference between two angles:
  • \(\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\)
To use this formula, substitute the angles you are working with into the equation. For example, for the problem \(\tan(x - \frac{\text{蟺}}{2})\), let \(\text{a} = x\) and \(\text{b} = \frac{\text{蟺}}{2}\). This easily transforms the initial equation to the desired form.
Cotangent
The cotangent function, denoted as \(\text{cot}\), is the reciprocal of the tangent function. If you know the value of \(\tan x\), you can easily find \(\text{cot} x\):
\(\text{cot} x = \frac{1}{\tan x}\)

Cotangent is particularly useful when transforming expressions involving tangent, especially during simplifications. In our example:
\(\tan\bigg(x - \frac{\text{蟺}}{2}\bigg) = -\frac{1}{\tan x} = -\text{cot} \big(x\big)\)

By recognizing the relationship between tangent and cotangent, you effectively simplify the expression and gain a deeper understanding of trigonometric functions and their relationships.

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