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Chapter 5: Problem 55

Use the sum-to-product formulas to rewrite the sum or difference as a product. $$\cos 6 x+\cos 2 x$$

Short Answer

Expert verified
The function $\cos 6x + \cos 2x$ can be rewritten as a product using the sum-to-product formula as $2 \cos(4x) \cos(2x)$.

Step by step solution

01

Identify the Values for \(\alpha\) and \(\beta\)

Looking at the given equation, identify the values for \(\alpha\) and \(\beta\). Here, \(\alpha = 6x\) and \(\beta = 2x\). You will plug these into the cos(伪 + 尾) and cos(伪 - 尾) formulas.
02

Plug in the values for \(\alpha\) and \(\beta\) to the Sum-to-Product Formula

Next, you need to plug the values for \(\alpha\) and \(\beta\) into the sum-to-product formula for \(\cos(\alpha) + \cos(\beta)\). The formula is:\[2 \cos \left( \frac{ \alpha + \beta }{2} \right) \cos \left( \frac{ \alpha - \beta }{2} \right)\]For \(\alpha = 6x\) and \(\beta = 2x\), the equation becomes: \[2 \cos \left( \frac{ 6x + 2x }{2} \right) \cos \left( \frac{ 6x - 2x }{2} \right)\]
03

Simplify the expression

The final step is to simplify the expression obtained in Step 2. This gives: \[2 \cos (4x) \cos(2x)\]

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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 useful tools in algebra and trigonometry. They help us transform expressions or equations using relationships between trigonometric functions. One such family of identities is the sum-to-product formulas. These are used to convert sums or differences of trigonometric functions into products. This can simplify calculations and make solving equations easier.
For example, the formula for the sum of two cosine functions is:
  • \( \cos(\alpha) + \cos(\beta) = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right) \)
Understanding these identities can help you recognize patterns and manipulate expressions for simplification or further calculation.
Cosine Function
The cosine function is one of the primary trigonometric functions, often abbreviated as \( \cos \). It represents the ratio of the adjacent side to the hypotenuse in a right triangle.
The cosine function is periodic, with a period of \( 2\pi \), meaning its values repeat every \( 2\pi \). When examining or manipulating trigonometric expressions, especially with cosine, it's important to remember its fundamental properties.
  • Symmetry: \( \cos(-x) = \cos(x) \)
  • Periodicity: \( \cos(x + 2\pi) = \cos(x) \)
  • Boundedness: \(-1 \leq \cos(x) \leq 1 \)
These properties help simplify expressions like \( \cos(6x) + \cos(2x) \) into more workable forms using identities like the sum-to-product formulas.
Algebraic Manipulation
Algebraic manipulation involves rearranging expressions or equations using mathematical operations and properties. Basic steps involve identifying relevant formulas, substituting values, and simplifying the results.
In the given problem, we applied the sum-to-product formula. Here's how:
  • First, identify \( \alpha = 6x \) and \( \beta = 2x \).
  • Then, substitute \( \alpha \) and \( \beta \) in the sum-to-product formula for cosines: \[\cos(\alpha) + \cos(\beta) = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right)\]
  • Perform the arithmetic within the cosine's argument: \[\frac{6x + 2x}{2} = 4x\] and \[\frac{6x - 2x}{2} = 2x\]
  • Simplify to get the final expression: \[2 \cos (4x) \cos(2x)\]
This manipulation turns a trigonometric sum into a product, simplifying the expression and making it easier to work with in further calculations.

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Most popular questions from this chapter

Fill in the blanks. When solving a trigonometric equation, the preliminary goal is to ________ the trigonometric function involved in the equation.

Determine whether the statement is true or false. Justify your answer. If you correctly solve a trigonometric equation to the statement \(\sin x=3.4,\) then you can finish solving the equation by using an inverse function.

Use the sum-to-product formulas to rewrite the sum or difference as a product. $$\sin 5 \theta-\sin 3 \theta$$

A weight is attached to a ,spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by $$y=\frac{1}{3} \sin 2 t+\frac{1}{4} \cos 2 t$$,where \(y\) is the distance from equilibrium (in feet) and \(t\) is the time (in seconds). (a) Use the identity \(a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \sin (B \theta+C)\) where \(C=\arctan (b / a), a>0,\) to write the model in the form \(y=\sqrt{a^{2}+b^{2}} \sin (B t+C)\). (b) Find the amplitude of the oscillations of the weight. (c) Find the frequency of the oscillations of the weight.

The height \(h\) (in feet) above ground of a seat on a Ferris wheel at time \(t\) (in minutes) can be modeled by $$h(t)=53+50 \sin \left(\frac{\pi}{16} t-\frac{\pi}{2}\right)$$. The wheel makes one revolution every 32 seconds. The ride begins when \(t=0\). (a) During the first 32 seconds of the ride, when will a person on the Ferris wheel be 53 feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 160 seconds, then how many times will a person be at the top of the ride, and at what times?
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