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Chapter 6: Problem 47

Use a graphing utility to check your result graphically. $$\frac{\cos ^{2} x-4}{\cos x-2}$$

Short Answer

Expert verified
The simplified version of the given expression is \(\cos x + 2\). Confirming this graphically will show the two functions overlap which means they are equivalent.

Step by step solution

01

Factor the expression

To simplify the given expression, one must first factor the numerator. Factoring gives \(\cos^2 x - 4 = (\cos x - 2)(\cos x + 2)\). So, the original expression can be rewritten as \(\frac{(\cos x - 2)(\cos x + 2)}{\cos x - 2}\).
02

Simplify the expression

Next, one notices that the term \(\cos x - 2\) is in both the numerator and the denominator of the fraction, which allows for the simplification of the expression by canceling out this factor. The resulting simplified expression is \(\cos x + 2\) after cancelling out.
03

Graphical Confirmation

Lastly, a graph is created of both the original expression and the simplified expression. These graphs should overlap, showing that the original and the simplified versions of the expression are indeed equivalent.

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Key Concepts

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

Cosine Function
Understanding the cosine function is paramount in simplifying trigonometric expressions. The cosine function, often expressed as cos(x), is a fundamental trigonometric function that relates the adjacent side to the hypotenuse in a right-angled triangle. It is also a periodic function, repeating its values at intervals of 2蟺 radians or 360掳.

When approaching problems involving cosine, remember it's an even function, meaning cos(鈭抶) = cos(x). This property is useful when simplifying expressions and can be seen in practice when graphing the function. The wave-like graph of cosine starts at a maximum value of 1 when x = 0 and smoothly transitions through its period.
Factoring Expressions

Essence of Factoring

Factoring expressions is a crucial algebraic technique used in simplifying trigonometric expressions, solving equations, and analyzing mathematical relationships. When factoring, one seeks to rewrite an expression as a product of its factors. These factors are simpler expressions that, when multiplied together, give back the original expression.

Factoring is akin to breaking down a complex problem into manageable parts. For instance, the difference of squares, a^2 - b^2, can be factored into (a + b)(a - b), making it simpler to handle, especially when part of a larger expression.
Graphing Utility

Verifying Equivalence Graphically

A graphing utility is a powerful tool for visualizing and confirming mathematical concepts, including the equivalence of trigonometric expressions. By inputting the original and simplified forms of the expression into the utility, one can generate their respective graphs. If the expressions are indeed equivalent, then their graphs will coincide.

Using such utilities not only helps in verifying the results but also provides a visual understanding of how trigonometric functions behave, fostering a deeper comprehension of the mathematical relationship depicted.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the included variables. They are the backbone of simplifying trigonometric expressions, providing tools to rewrite expressions in different forms. Common identities include Pythagorean identities, angle sum and difference identities, and double-angle identities.

In this instance, we leveraged the difference of squares identity to factor the given expression. Recognizing and applying these identities appropriately can transform a complicated expression into a simpler, more manageable one, aiding in both understanding and computation.

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Most popular questions from this chapter

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment connecting the points. $$\left(-1, \frac{1}{2}\right),\left(\frac{4}{3}, \frac{5}{2}\right)$$

Rewrite the expression in terms of \(\sin \theta\) and \(\cos \theta\) $$\frac{\csc \theta(1+\cot \theta)}{\tan \theta+\cot \theta}$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\cos ^{4} x-\sin ^{4} x=\cos 2 x$$

Find the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\) Use a graphing utility to verify your answers. $$\sin ^{2} 3 x-\sin ^{2} x=0$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$165^{\circ}$$
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