Identify the amplitude and period of the following functions. g ( θ ) = 3 cos ( θ 3 ) g\left(\theta\right)=3\cos\left(\frac{\theta}{3}\right) g ( θ ) = 3 cos ( 3 θ )

Welcome back, everyone. In this problem, we want to find the amplitude and period for the trigonometric expression P of X equals 5 multiplied by the cosine of a fourth of X for our answer choices. A says the amplitude is 5 and the period is a 4th of pie. B says our amplitude is 5 and our period is 8 pie. C says the amplitude is a 4th and the period is 5 pi. And the D says the amplitude is 4 and the period is a 5th of pi. Now, what do we want to find here? We want to find the amplitude and the period for our trigonometric expression. Recall that for a trigonometric function, they're generally written in the form a multiplied by the trig function. In this case, the cosine of b x minus c plus d, where our amplitude equals a and the period equals 2pi divided by b. So what this tells us is that if we can figure out the values of a and b from our trigonometric expression, then we should be able to find the amplitude and the period. So let's do that. Firstly, notice that a is the coefficient of the trigonometric term. So, in this case, a would be equal to 5. Additionally, B is the coefficient of the X term, so that means since our term is a fourth of X, then our coefficient of B would be 1 fourth. Now that we have the values of A and B, we can find our amplitude, which would be equal to A. So in this case, our amplitude is 5 and our period in this case, it would be 2 pi divided by a 4th, which we can rewrite as 2 pi multiplied by 4 divided by 1. And 2 pi multiplied by 4 is 8 pi. Therefore, our period is 8pi and our amplitude is 5 And when we look on our answer choices, the correct choice would have been answer choice b. Thanks a lot for watching everyone. I hope this video helped.

0. Functions

Graphs of Trigonometric Functions

Calculus

0. Functions

Graphs of Trigonometric Functions

2:40 minutes

Problem 97

Textbook Question

Identify the amplitude and period of the following functions.
$g\left(\theta\right)=3\cos\left(\frac{\theta}{3}\right)$

Verified step by step guidance

The given function is g(θ) = 3cos(θ/3). This is a cosine function of the form a * cos(bθ), where a is the amplitude and b affects the period.

Identify the amplitude: In the function g(θ) = 3cos(θ/3), the coefficient of the cosine function is 3. Therefore, the amplitude is the absolute value of this coefficient, which is |3| = 3.

Determine the period: The period of a basic cosine function cos(θ) is 2π. For a function of the form cos(bθ), the period is adjusted by the factor b, and is given by the formula 2π/|b|.

In the function g(θ) = 3cos(θ/3), the value of b is 1/3 (since θ/3 can be rewritten as (1/3)θ). Therefore, the period is 2π divided by 1/3, which simplifies to 2π * 3.

Thus, the amplitude of the function is 3, and the period is 6π.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Amplitude

Amplitude refers to the maximum value of a periodic function, specifically the height of the wave from its midline to its peak. In the function g(θ) = 3cos(θ/3), the amplitude is represented by the coefficient of the cosine function, which is 3. This means the function oscillates between 3 and -3.

Period

The period of a periodic function is the length of one complete cycle of the wave. For the cosine function, the standard period is 2π. In the function g(θ) = 3cos(θ/3), the period is affected by the coefficient of θ inside the cosine. The period can be calculated using the formula 2π divided by the coefficient of θ, which in this case is 1/3, resulting in a period of 6π.

Phase Shift

Phase shift refers to the horizontal shift of a periodic function along the x-axis. While the given function g(θ) = 3cos(θ/3) does not include a phase shift term, understanding this concept is essential for analyzing more complex functions. A phase shift occurs when the argument of the cosine function is adjusted by a constant, affecting where the wave starts on the x-axis.

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