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Chapter 6: Problem 39

Factor each trigonometric expression. $$\sin \alpha \cos \alpha+\cos \alpha+\sin \alpha+1$$

Short Answer

Expert verified
The factored form is \(\(\cos \alpha + 1\)(\sin \alpha + 1)\).

Step by step solution

01

Identify common factors

First, observe the given expression \(\sin \alpha \cos \alpha + \cos \alpha + \sin \alpha + 1\). Look for common factors in pairs or groups.
02

Group the terms

Group the terms in pairs: \(\sin \alpha \cos \alpha + \cos \alpha\) and \(\sin \alpha + 1\).
03

Factor out common terms in each group

From the first group, \(\sin \alpha \cos \alpha + \cos \alpha\), factor out \(\cos \alpha\). From the second group, \(\sin \alpha + 1\), factor out \(1\). This results in: \(\cos \alpha (\sin \alpha + 1) + 1(\sin \alpha + 1)\).
04

Factor out the common binomial factor

Notice that \(\sin \alpha + 1\) is a common factor in both terms. Factor \(\sin \alpha + 1\) out: \(\(\cos \alpha + 1\)(\sin \alpha + 1)\).

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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 fundamental tools in trigonometry. They are equations that hold true for all values during which the functions are defined. Some common trigonometric identities include the Pythagorean identities, such as \( \sin^2\theta + \cos^2\theta = 1\) and angle sum identities, like \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta\).
Understanding these identities allows for simplification and factoring of more complex trigonometric expressions.
In our example, although we did not directly apply a specific identity, knowing them provides a strong foundational understanding as we manipulated and grouped the terms.
common factors
Identifying common factors in an expression is a key step in factoring. A common factor is a term that appears in every part of the expression. In our example, we started by observing the given expression: \(\sin\alpha\cos\alpha + \cos\alpha + \sin\alpha + 1\).
We grouped the terms and identified \(\cos\alpha\) in the first pair and \(1\) in the second pair as common factors.
This technique helps to simplify and eventually factor out more complex expressions by reducing them to more manageable parts.
grouping terms
Grouping is a technique used to simplify an expression by organizing terms into pairs or sets that share a common factor. This makes the process of factoring simpler.
In the expression \(\sin\alpha\cos\alpha + \cos\alpha + \sin\alpha + 1\), we grouped terms into \(\sin\alpha\cos\alpha + \cos\alpha\) and \(\sin\alpha + 1\).
By organizing the terms this way, we could then factor out the common terms in each group. This approach often reveals patterns or factors that are not immediately obvious.
binomial factorization
Binomial factorization involves expressing a polynomial as the product of binomials. In our final step, we used this method to completely factor the expression.
After grouping and factoring out the common terms, we recognized \(\sin\alpha + 1\) as a binomial common to both groups.
Finally, we factored the expression to \( (\cos\alpha + 1)(\sin\alpha + 1) \), neatly separating it into two binomials.
This approach not only simplifies the polynomial but also often reveals meaningful mathematical properties.

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Most popular questions from this chapter

Find all values of \(\theta\) in the interval \(0^{\circ}, 360^{\circ}\) ) that satisfy each \right. equation. Round approximate answers to the nearest tenth of a degree. $$2 \sin ^{2}\left(\frac{\theta}{2}\right)=\cos \theta$$

Solve each problem. Wave Action The vertical position of a floating ball in an experimental wave tank is given by the equation \(x=2 \sin (\pi t / 3)\) where \(x\) is the number of feet above sea level and \(t\) is the time in seconds. For what values of \(t\) is the ball \(\sqrt{3} \mathrm{ft}\) above sea level?

Verify that each equation is an identity. $$\frac{1-\sin ^{2}\left(\frac{x}{2}\right)}{1+\sin ^{2}\left(\frac{x}{2}\right)}=\frac{1+\cos x}{3-\cos x}$$

Verify that each equation is an identity. $$(\sin \alpha-\cos \alpha)^{2}=1-\sin 2 \alpha$$

If \(\beta\) is an angle in standard position such that \(\sin (\beta)=1\) then what is \(\cos (\beta) ?\)
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