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Chapter 20: Problem 14

Multiply and simplify. $$\sec \theta+\sec ^{3} \theta$$

Short Answer

Expert verified
The simplified expression is \( \sec \theta (2 + \tan^2 \theta) \).

Step by step solution

01

Factor Out the Common Term

First, observe the expression \( \sec \theta + \sec^3 \theta \). The common factor here is \( \sec \theta \). Factoring \( \sec \theta \) out of the expression gives us: \[ \sec \theta (1 + \sec^2 \theta). \]
02

Simplify Using Trigonometric Identities

We know from trigonometric identities that \( \sec^2 \theta = 1 + \tan^2 \theta \). Substitute this identity into the expression obtained in Step 1:\[ \sec \theta (1 + 1 + \tan^2 \theta). \]This becomes:\[ \sec \theta (2 + \tan^2 \theta). \]

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Key Concepts

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

Factoring Techniques
Factoring is a powerful technique in mathematics, especially in simplifying algebraic expressions. In this exercise, the expression given is \( \sec \theta + \sec^3 \theta \). The first step is to identify a common factor, which in this case is \( \sec \theta \).

When you factor out \( \sec \theta \), you can simplify the expression as follows:
  • Look for terms that can be pulled out because they appear in every term of the expression.
  • In our example, \( \sec \theta \) is the factor shared by both terms.

By factoring out \( \sec \theta \), the expression becomes \( \sec \theta (1 + \sec^2 \theta) \).

Factoring simplifies expressions by making them easier to work with and is an essential skill for solving complex mathematical problems.
Mathematical Simplification
Simplifying mathematical expressions often involves using known identities or properties to reduce complexity. In the given task, we use the trigonometric identity, \( \sec^2 \theta = 1 + \tan^2 \theta \), to further simplify the expression.
  • Recognize and apply mathematical identities to transform expressions.
  • Substitute known identities, like changing \( \sec^2 \theta \) into \( 1 + \tan^2 \theta \), to simplify the problem.

Replacing \( \sec^2 \theta \) with \( 1 + \tan^2 \theta \), our expression becomes \( \sec \theta (2 + \tan^2 \theta) \). This step not only simplifies calculations but also helps in understanding the relationships among trigonometric functions more deeply.
Secant Function
The secant function is a fundamental part of trigonometry. It's the reciprocal of the cosine function. The nomenclature \( \sec \theta \) is actually \( \frac{1}{\cos \theta} \). This implies that it represents the ratio of the hypotenuse to the adjacent side in a right triangle.

Here are some key aspects about the secant function:
  • Domain: The function is not defined wherever \( \cos \theta = 0 \), such as \( \theta = 90^\circ, 270^\circ \), etc.
  • Range: It includes all values except for the interval \((-1, 1)\).
  • Graph: The secant graph has vertical asymptotes where the cosine graph crosses the x-axis.

In this task, understanding the properties of \( \sec \theta \) aids in simplifying the initial expression. Its interactions with \( \tan \theta \) through identities highlight integral trigonometric principles useful in various math and physics applications.

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Most popular questions from this chapter

Evaluate the given expressions. $$\sin ^{-1} x+\cos ^{-1} x$$

Solve the given problems. The cross section of a radio-wave reflector is defined by \(x=\cos 2 \theta, y=\sin \theta .\) Find the relation between \(x\) and \(y\) by eliminating \(\theta\)

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Solve the indicated equations analytically. Solve the system of equations \(r=\sin \theta, \quad r=\sin 2 \theta,\) for \(0 \leq \theta<2 \pi\)

Solve the given problems. Show that \(\sin ^{2} x\left(1-\sec ^{2} x\right)+\cos ^{2} x\left(1+\sec ^{4} x\right)\) has a con- stant value.
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