Problem 67 Complete. (For example, \(\sin (... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 6: Problem 67

Complete. (For example, $\sin (x+2 \pi)=\sin x$ ) $$\cos (x-\pi)=$$

Short Answer

Expert verified

$\cos (x - \pi) = -\cos (x)$

Step by step solution

Recall the Trigonometric Identity

Recall the trigonometric identity for cosine, which states \ \ \[\cos(x - \pi) = -\cos(x)\]. This identity is derived from the fact that cosine is an even function and \ \ \[\cos(\theta + \pi) = -\cos(\theta)\].

Apply the Identity

Apply the cosine identity to the given expression. Substituting \ \ \[ \theta = x - \pi \] results in \ \ \[\cos(x - \pi) = -\cos(x)\].

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Key Concepts

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

Cosine function

The cosine function is one of the fundamental functions in trigonometry. It is often denoted as $\text{cos}(x)$. The cosine function is used to relate the angles of a triangle to the lengths of its sides in a right-angled triangle.
Some important properties of the cosine function include:

Range: The values of the cosine function lie between -1 and 1 i.e., $-1 \leq \cos(x) \leq 1$.
Periodicity: The cosine function has a period of $2\pi$, meaning $\text{cos}(x + 2\pi) = \text{cos}(x)$.
Symmetry: The cosine function is symmetric with respect to the y-axis.

The graph of the cosine function is a wave that oscillates between -1 and 1.
It starts from 1 when $x = 0$ and reaches -1 at $\text{cos}(\pi)$. The function repeats this pattern every $2\pi$ units. Understanding these properties helps in solving various trigonometric identities and transformations.

Even function

An even function is a type of function that satisfies the rule $f(-x) = f(x)$. This means the function produces the same output for both $+x$ and $-x$.
For example,

For the cosine function, $\text{cos}(-x) = \text{cos}(x)$ shows it's an even function.
Graphically, even functions are symmetric with respect to the y-axis.

We can use this property to simplify trigonometric expressions and solve equations. Given the even nature of cosine, the identity $\text{cos}(x - \pi) = -\text{cos}(x)$ is derived. This property is particularly useful when dealing with transformations involving shifts and reflections.

Cosine transformation

Cosine transformations involve modifying the argument of the cosine function to achieve shifts, reflections, and stretches.
Key transformations include:

Horizontal Shifts: To shift the cosine graph horizontally, we add or subtract a value inside the argument. For example, $\text{cos}(x - \pi)$ represents a shift to the right by $\pi$ units.
Vertical Shifts: To shift the graph vertically, we add or subtract a value outside the function. For instance, $\text{cos}(x) + c$ shifts the graph up by $c$ units.
Reflections: The graph can be reflected across the x-axis by multiplying the cosine function by -1, i.e., $-\text{cos}(x)$.
Scaling: To stretch or compress the graph horizontally, we multiply the argument by a constant. For example, $\text{cos}(bx)$ compresses the graph if $b > 1$ and stretches it if $0 < b < 1$.

In the given problem, $\text{cos}(x - \pi)$, we identify this as a horizontal shift and reflection combination. This transformation is a crucial tool in trigonometry for graphing and solving equations efficiently.

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91影视

Complete. (For example, \(\sin (x+2 \pi)=\sin x\) ) $$\cos (x-\pi)=$$

Short Answer

Step by step solution

Recall the Trigonometric Identity

Apply the Identity

Key Concepts

Cosine function

Even function

Cosine transformation

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