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Write an equation of the cosine function with amplitude 2 and period [tex]$6 \pi$[/tex].

A. [tex]y = -2 \cos \left(\frac{1}{6} x \right)[/tex]
B. [tex]y = -\frac{1}{2} \cos \left(\frac{1}{3} x \right)[/tex]
C. [tex]y = 2 \cos \left(\frac{1}{3} x \right)[/tex]
D. [tex]y = \frac{1}{2} \cos \left(\frac{1}{6} x \right)[/tex]

Sagot :

GinnyAnsweridn
Let's solve the problem step-by-step:

### Step 1: Understanding amplitude
The amplitude of a cosine function is the coefficient in front of the cosine. For a general function of the form [tex]\( y = A \cos(Bx) \)[/tex], the amplitude is [tex]\(|A|\)[/tex].

Given the amplitude is 2, we set [tex]\( |A| = 2 \)[/tex]. Therefore, [tex]\( A \)[/tex] can be either [tex]\( +2 \)[/tex] or [tex]\(-2\)[/tex].

### Step 2: Understanding the period
The period of the cosine function is determined by the coefficient [tex]\( B \)[/tex] inside the argument of the cosine function. For a general cosine function of the form [tex]\( y = A \cos(Bx) \)[/tex], the period [tex]\( T \)[/tex] is given by:

[tex]\[ T = \frac{2\pi}{B} \][/tex]

We are given that the period is [tex]\( 6\pi \)[/tex]. So we set up the equation:

[tex]\[ \frac{2\pi}{B} = 6\pi \][/tex]

### Step 3: Solving for [tex]\( B \)[/tex]
To find [tex]\( B \)[/tex], we solve the equation:

[tex]\[ \frac{2\pi}{B} = 6\pi \][/tex]

Multiplying both sides by [tex]\( B \)[/tex] and then dividing by [tex]\(6\pi \)[/tex], we get:

[tex]\[ B = \frac{2\pi}{6\pi} = \frac{1}{3} \][/tex]

### Step 4: Forming the equation
Using the values we have found for [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:

1. Amplitude ([tex]\( A \)[/tex]) = 2
2. [tex]\( B = \frac{1}{3} \)[/tex]

The general form of the equation is:

[tex]\[ y = 2 \cos \left( \frac{1}{3} x \right) \][/tex]

### Step 5: Checking the options
Let's compare this with the given multiple-choice options:

1. [tex]\( y = -2 \cos \left( \frac{1}{6} x \right) \)[/tex]
2. [tex]\( y = -\frac{1}{2} \cos \left( \frac{1}{3} x \right) \)[/tex]
3. [tex]\( y = 2 \cos \left( \frac{1}{3} x \right) \)[/tex]
4. [tex]\( y = \frac{1}{2} \cos \left( \frac{1}{6} x \right) \)[/tex]

The correct option that matches our derived equation is:

[tex]\[ y = 2 \cos \left( \frac{1}{3} x \right) \][/tex]

Thus, the equation of the cosine function with amplitude 2 and period [tex]\( 6 \pi \)[/tex] is:

[tex]\[ y = 2 \cos \left( \frac{1}{3} x \right) \][/tex]

So, the correct answer is:
[tex]\[ y = 2 \cos \left( \frac{1}{3} x \right) \][/tex]
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