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Chapter 7: Problem 56

Use a graphing calculator to determine which expression \((A)-(F)\) on the right can be used to complete the identity. Then try to prove that identity algebraically. $$(\cos x+\sin x)(1-\sin x \cos x)$$ A. \(\frac{\sin ^{3} x-\cos ^{3} x}{\sin x-\cos x}\) B. \(\cos x\) C. \(\tan x+\cot x\) D. \(\cos ^{3} x+\sin ^{3} x\) E. \(\frac{\sin x}{1-\cos x}\) F. \(\cos ^{4} x-\sin ^{4} x\)

Short Answer

Expert verified
The answer is B: \ \cos x \.

Step by step solution

01

Expand the Left-Hand Side (LHS)

First, expand the expression \( (\cos x + \sin x)(1 - \sin x \cos x) \) using the distributive property. \ (\cos x + \sin x)(1 - \sin x \cos x) = (\cos x + \sin x) - (\cos x + \sin x) \sin x \cos x \
02

Simplify the Expanded Expression

Next, combine like terms in the expanded expression: \(\cos x + \sin x - \cos x \sin x \cos x - \sin x \sin x \cos x\). This simplifies to \( \cos x + \sin x - \sin x \cos^2 x - \sin^2 x \cos x \).
03

Factor the Expression

Now, factor out common terms. The expression \( \cos x + \sin x - \sin x \cos^2 x - \sin^2 x \cos x \) can be rewritten by factoring terms containing \ \sin x \ and \ \cos x \.
04

Use Trigonometric Identities

Recognize that \( \cos^2 x \sin x + \sin^2 x \cos x \) can be simplified further using trigonometric identities such as \sin^2 x + \cos^2 x = 1\. Apply this identity to the expression.
05

Compare with Given Options

Compare the simplified expression with each of the provided options: A. \( \frac{\sin^3 x - \cos^3 x}{\sin x - \cos x} \) B. \( \cos x \) C. \( \tan x + \cot x \) D. \( \cos^3 x + \sin^3 x \) E. \( \frac{\sin x}{1 - \cos x} \) F. \( \cos^4 x - \sin^4 x \) The simplified expression resembles option B the most closely.
06

Prove Algebraically

To prove the solution algebraically, show that \( \cos x + \sin x - \sin x \cos^2 x - \sin^2 x \cos x \) simplifies to \( \cos x \). Since all other terms cancel each other out, the remaining term is \( \cos x \).

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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 that involve trigonometric functions and are true for every value of the included variables. Four essential identities often used in simplifying expressions are:
  • Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
  • Reciprocal Identities: \(\sin x = 1/\text{csc} x \) and \(\cos x = 1/\text{sec} x \)
  • Co-Function Identities: \(\sin ( \frac{\text{蟺}}{2} - x) = \cos x \) and \(\cos ( \frac{\text{蟺}}{2} - x) = \sin x \)
  • Even-Odd Identities: \(\sin (-x) = -\text{sin} x \) and \(\cos (-x) = \cos x \)
Using these identities in our example helped us simplify the keys terms such as \(\cos^2 x + \sin^2 x) = 1 \). This simplification is powerful and allows us to see the resemblance to \(\cos x \).
Algebraic Simplification
Algebraic simplification involves reducing expressions to a simpler form. In our exercise, we expanded and simplified our initial expression step-by-step:
  • First, we used the distributive property:
    \((\text{cos} x + \text{sin} x)(1 - \text{sin} x \text{cos} x) = \text{cos} x + \text{sin} x - (\text{cos} x + \text{sin} x) \text{sin} x \text{cos} x\)
  • Next, we handled similar terms:
    \(\text{cos} x + \text{sin} x - \text{cos} x \text{sin} x \text{cos} x \) and \(\text{cos} x + \text{sin} x - \text{sin}^2 x \text{cos} x \)
  • By recognizing common trigonometric forms, simplifying becomes straightforward.
This approach, like employing trigonometric identities, is powerful in breaking down complex expressions into understandable parts.
Graphing Calculator Usage
Graphing calculators are valuable tools in verifying solutions and exploring expressions. Here鈥檚 how to use a graphing calculator for trigonometry:
  • Enter the expanded expression and the given options one by one.
  • Graph the functions to see how their behaviors compare across various values of \( x \).
  • Check for matches. When the behavior of a graphed option mirrors the original expression, you have a match.
In this exercise, entering \( (\text{cos} x + \text{sin} x)(1 - \text{sin} x \text{cos} x)\) and graphing all options A to F side-by-side shows that expression B \( \text{cos} x \) directly overlaps, proving our expansion matches \( \text{cos} x \).
Right Triangle Trigonometry
Understanding the relationships within right triangles is fundamental to trigonometry. Key aspects include:
  • The Opposite Side: the side across from the angle of interest.
  • The Adjacent Side: the side next to the angle of interest.
  • The Hypotenuse: the longest side, opposite the right angle.
The primary trigonometric functions are defined as:
  • \( \text{sin} x = \frac{\text{\text{Opposite Side}}}{\text{Hypotenuse}} \)
  • \( \text{cos} x = \frac{\text{\text{Adjacent Side}}}{\text{Hypotenuse}} \)
  • \( \text{tan} x = \frac{\text{\text{Opposite Side}}}{\text{Adjacent Side}} \)
These definitions form the basis for more complex identities and simplifications, serving as the groundwork for problem-solving and understanding trigonometric functions in full extent.

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Most popular questions from this chapter

Electrical Theory. In electrical theory, the following equations occur: $$E_{1}=\sqrt{2} E_{t} \cos \left(\theta+\frac{\pi}{P}\right)$$ and $$E_{2}=\sqrt{2} E_{t} \cos \left(\theta-\frac{\pi}{P}\right)$$. Assuming that these equations hold, show that $$\frac{E_{1}+E_{2}}{2}=\sqrt{2} E_{t} \cos \theta \cos \frac{\pi}{P}$$ and $$\frac{E_{1}-E_{2}}{2}=-\sqrt{2} E_{t} \sin \theta \sin \frac{\pi}{P}$$.

Consider the following functions ( \(a\) )- ( \(f\) ). Without graphing them, answer question. a) \(f(x)=2 \sin \left(\frac{1}{2} x-\frac{\pi}{2}\right)\) b) \(f(x)=\frac{1}{2} \cos \left(2 x-\frac{\pi}{4}\right)+2\) c) \(f(x)=-\sin \left[2\left(x-\frac{\pi}{2}\right)\right]+2\) d \(f(x)=\sin (x+\pi)-\frac{1}{2}\) e) \(f(x)=-2 \cos (4 x-\pi)\) f) \((x)=-\cos \left[2\left(x-\frac{\pi}{8}\right)\right]\) Which functions have a graph with a phase shift of \(\frac{\pi}{4} ?\)

Simplify. $$\cos (u+v) \cos v+\sin (u+v) \sin v$$

First write each of the following as a trigonometric function of a single angle. Then evaluate. $$\cos 83^{\circ} \cos 53^{\circ}+\sin 83^{\circ} \sin 53^{\circ}$$

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { linear speed } & \text { congruent } \\ \text { angular speed } & \text { circular } \\ \text { angle of elevation } & \text { periodic } \\ \text { angle of depression } & \text { period } \\ \text { complementary } & \text { amplitude } \\ \text { supplementary } & \text { quadrantal } \\ \text { similar } & \text { radian measure }\end{array}$$ Triangles are ________________ if their corresponding angles have the same measure.
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