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Chapter 5: Problem 55

Graph each of the functions by first rewriting it as a sine, cosine, or tangent of a difference or sum. $$ y=-\sin x \cos (3 x)-\cos x \sin (3 x) $$

Short Answer

Expert verified
The function simplifies to \( y = -\sin(4x) \), a sine wave with 4 oscillations per \(2\pi\).

Step by step solution

01

Identify Trigonometric Identity

Recognize that the given function resembles the formula for the sine of the sum of angles, which is \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \].
02

Rewrite Using Identity

The given function is \[ y = -\sin x \cos(3x) - \cos x \sin(3x) \]. Using the identity, we can rewrite this as \[ y = -\sin(x + 3x) \].
03

Simplify the Equation

Simplify \[ y = -\sin(x + 3x) \] to get \[ y = -\sin(4x) \].
04

Graph the Function

Graph \[ y = -\sin(4x) \], which is a sine function with:- Amplitude: 1- Period: \[ \frac{2\pi}{4} = \frac{\pi}{2} \]- Phase Shift: None- Vertical Shift: NonePlotting the sine wave will have the peaks at -1 and 1 with four oscillations between 0 and \(2\pi\).

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Key Concepts

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

Graphing Trigonometric Functions
Graphing trigonometric functions can initially seem like a complex task, but it's a skill that becomes easier with practice. Trigonometric functions like sine, cosine, and tangent have specific shapes and characteristics that are identifiable once you understand their basic components.

To graph a function, identify its amplitude, period, phase shift, and vertical shift. Amplitude refers to the height of the wave, while the period measures the length of one complete cycle of the wave. The phase shift indicates if the wave starts at a point other than the origin, and a vertical shift moves the entire graph up or down.

In our exercise, we graph the function \( y = -\sin(4x) \). This function is a reflection of the sine wave, due to the negative sign, and is compressed horizontally. Understanding these transformations is key to accurately graphing trigonometric functions.
Sine of Sum Identity
The sine of sum identity is a useful trigonometric tool that simplifies expressions involving the sine of an angle sum. It is given by the formula: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \].

In our given exercise, this identity helps to rewrite a complex expression into a simpler form. Initially, we have \( y = -\sin x \cos(3x) - \cos x \sin(3x) \). By recognizing this structure as the sine of sum identity, you can see that it becomes \( y = -\sin(x + 3x) \).

Simplification of such expressions using identities is crucial as it makes graphing and further calculations much more straightforward. In this case, it results in the expression \( y = -\sin(4x) \), which is much easier to handle.
Amplitude and Period of Trig Functions
When analyzing trigonometric functions, amplitude and period are principal components determining the shape of the graph.

The amplitude of a trigonometric function is the distance from the midline of the wave to its peak or trough. In basic sine and cosine functions, this distance is 1; however, coefficients can change it. In \( y = -\sin(4x) \), the amplitude is 1, but the wave is reflected because of the negative sign.

The period defines how long it takes for the function to repeat its pattern. It's given by the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the function. For \( y = -\sin(4x) \), the period is \( \frac{\pi}{2} \), indicating that the function completes four full cycles between 0 and \( 2\pi \).

Understanding these attributes allows for precise graphing and analysis of trigonometric functions.

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Most popular questions from this chapter

Consider the triangle below, where the vertex angle measures \(\theta\), the equal sides measure \(a\), the height is \(h\), and half the base is \(b\). (In an isosceles triangle, the perpendicular dropped from the vertex angle divides the triangle into two congruent triangles.) The two triangles formed are right triangles. In the right triangles, \(\sin \left(\frac{\theta}{2}\right)=\frac{b}{a}\) and \(\cos \left(\frac{\theta}{2}\right)=\frac{h}{a}\). Multiply each side of each equation by \(a\) to get \(b=a \sin \left(\frac{\theta}{2}\right), h=a \cos \left(\frac{\theta}{2}\right)\). The area of the entire isosceles triangle is \(A=\frac{1}{2}(2 b) h=b h\). Substitute the values for \(b\) and \(h\) into the area formula. Show that the area is equivalent to \(\frac{a^{2}}{2} \sin \theta\).

Let \(n\) be a positive integer. Express \(\sin \left(\frac{n \pi}{2}\right) \cos \left(\frac{n \pi}{2}\right)\) as a single trigonometric function, and then evaluate if possible.

Verify the identities. $$ \sec ^{2}\left(\frac{A}{2}\right)=\frac{2(1-\cos A)}{\sin ^{2} A} $$

Find the exact value of \(\sin 15^{\circ}\) in two ways, using sum and difference identities and half-angle identities; then show that they are equal.

For Exercises 55 and 56, refer to the following: Monthly profits can be expressed as a function of sales, that is, \(p(s)\). A financial analysis of a company has determined that the sales \(s\) in thousands of dollars are also related to monthly profits \(p\) in thousands of dollars by the relationship: $$ \tan \theta=\frac{p}{s} \text { for } 0 \leq s \leq 50,0 \leq p<40 $$ Based on sales and profits, it can be determined that the domain for angle \(\theta\) is \(0 \leq \theta \leq 38^{\circ}\). If monthly profits are $$\$ 3000$$ and monthly sales are $$\$ 4000$$, find \(\tan \left(\frac{\theta}{2}\right)\).
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