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Chapter 7: Problem 18

Factor each trigonometric expression. $$4 \tan ^{2} \beta+\tan \beta-3$$

Short Answer

Expert verified
\((4 \tan \beta - 3)(\tan \beta + 1)\)

Step by step solution

01

Identify the Quadratic Form

The given expression is in the form of a quadratic equation: \(4 \tan^{2} \beta + \tan \beta - 3\). Identify the corresponding terms: \(a = 4\), \(b = 1\), and \(c = -3\).
02

Set Up for Factoring

To factor the quadratic expression, we need to find two numbers that multiply to \(a \times c = 4 \times -3 = -12\) and add up to \(b = 1\).
03

Find Factor Pairs

The pairs of numbers that multiply to -12 and add up to 1 are 4 and -3. Thus, we rewrite the expression: \(4 \tan^{2} \beta + 4 \tan \beta - 3 \tan \beta - 3\).
04

Group Terms

Group the terms in pairs: \((4 \tan^{2} \beta + 4 \tan \beta) + (-3 \tan \beta - 3)\).
05

Factor by Grouping

Factor out the greatest common factor (GCF) from each group: \(4 \tan \beta(\tan \beta + 1) - 3(\tan \beta + 1)\).
06

Factor the Common Binomial

Notice that \(\tan \beta + 1\) is a common factor, so factor it out: \((4 \tan \beta - 3)(\tan \beta + 1)\).
07

Conclusion

The original expression \(4 \tan^{2} \beta + \tan \beta - 3\) is factored into \((4 \tan \beta - 3)(\tan \beta + 1)\).

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Key Concepts

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

quadratic equations
Quadratic equations are polynomial equations of the form \[ax^2 + bx + c = 0\]. In the given exercise, the quadratic equation is presented with a trigonometric function, \(\tan \beta\) in place of the variable \(x\), resulting in \[4 \tan^{2} \beta + \tan \beta - 3\].

To solve this, you identify the values of \(a\), \(b\), and \(c\), which are respectively 4, 1, and -3.
  • \(a\) represents the coefficient of the squared term \(\tan^2\beta\)
  • \(b\) represents the coefficient of the linear term \(\tan\beta\)
  • \(c\) represents the constant term
Quadratics can reveal important properties of functions and are foundational in algebra and calculus. Understanding how to factor them is crucial for solving more complex equations.
trigonometric functions
Trigonometric functions, such as \(\tan (\tan)\), \(\sin (\sin)\), and \(\cos (\cos)\), are fundamental in mathematics, describing relationships in triangles and modeling periodic phenomena. In this problem, the focus is on the tangent function \(\tan\).

The expression \(\tan^{2} \beta\) means \((\tan\beta)^2\), indicating it's squared. Factoring expressions involving trigonometric functions often involves viewing them as algebraic terms. This allows the use of algebraic techniques such as factoring quadratics, even though the functions themselves describe angles and ratios in trigonometry. Understanding these functions is key in many fields including physics and engineering.
factoring techniques
Factoring techniques are essential tools in algebra that simplify expressions and solve equations. In the given exercise, the goal is to factor the quadratic expression \[4 \tan^{2} \beta + \tan \beta - 3\]. Follow these steps:

  • Firstly, identify pairs of numbers that multiply to \( a \times c = 4 \times (-3) = -12 \) and add to \( b = 1 \)
  • The pairs are 4 and -3, since \( 4 \times (-3) = -12 \) and \( 4 + (-3) = 1 \)
  • Rewrite the original expression: \[4 \tan^{2} \beta + 4 \tan \beta - 3\tan \beta - 3\]
  • Group terms: \[(4 \tan\beta (\tan\beta + 1)) - (3 (\tan\beta + 1))\]
Notice \(\tan \beta + 1\) is a common factor, thus it can be factored out to get \[(4 \tan\beta - 3)(\tan\beta + 1)\]. These techniques involve recognizing common patterns and breaking down more complex expressions into simpler, product forms.

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Most popular questions from this chapter

The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval \([0,2 \pi)\). Express solutions to four decimal places. $$2 \sin 2 x-x^{3}+1=0$$

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Use the given information to find \(\cos (s+t)\) and \(\cos (s-t)\). $$\cos s=-\frac{1}{5} \text { and } \sin t=\frac{3}{5}, s \text { and } t \text { in quadrant II }$$
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