Identify the amplitude and period of the following functions. f ( π ) = 2 sin 2 θ f\left(\pi\right)=2\sin2\theta f ( π ) = 2 sin 2 θ

Welcome back, everyone. In this problem, we want to find the amplitude and period for the trigonometric expression. P of x equals 3 multiplied by the sign of 4 x. For our answer choices, A says the amplitude is 3 and the period is 4. B says the amplitude is 3 and the period is a half of 5. C says the amplitude is 3 and the period is a quarter of 5. And d says the amplitude is 4 and the period is a 3rd of pi. Now, what do we want to find here? We want to figure out the period and the amplitude for trigonometric expression. And recall that every trigonometric expression is usually written in the form a multiplied by the expression. In this case, the sign of b x minus c plus d. Where our amplitude equals a. And our period equals 2pi divided by b. So what this is telling us is that if we can figure out the values for a and b, we can substitute them to find our amplitude and our period. Now what do we know? Well, notice that a is the coefficient of the trigonometric term. So in this case, a would be equal to 3. And notice that b is the coefficient of the x term. So in this case, B would be equal to 4. So that tells us then that our amplitude. Would be equal to 3 since our amplitude is a and our period would be equal to 2 pi divided by 4. Notice both the numerator and the denominator have a common factor of 2. So by factoring 2 out, 2 goes into 2 one time, 2 into 4 two times. So that leaves us with a half of Pi. Therefore, our amplitude is 3 and our period is a half of Pi And when we look in our answer choices, that would have been answer choice b. Thanks a lot for watching everyone. I hope this video helped.

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0. Functions

Graphs of Trigonometric Functions

Calculus

0. Functions

Graphs of Trigonometric Functions

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Problem 96

Textbook Question

Identify the amplitude and period of the following functions.
$f\left(\pi\right)=2\sin2\theta$

Verified step by step guidance

Identify the general form of the sine function, which is $ f(\theta) = a \sin(b\theta) $, where $ a $ is the amplitude and $ \frac{2\pi}{b} $ is the period.

In the given function $ f(\theta) = 2\sin(2\theta) $, compare it with the general form to identify the values of $ a $ and $ b $. Here, $ a = 2 $ and $ b = 2 $.

The amplitude of a sine function is the absolute value of $ a $. Therefore, the amplitude is $ |2| = 2 $.

The period of a sine function is calculated using the formula $ \frac{2\pi}{b} $. Substitute $ b = 2 $ into the formula to find the period.

Calculate the period: $ \frac{2\pi}{2} = \pi $. Thus, the period of the function is $ \pi $.

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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, particularly in trigonometric functions like sine and cosine. It indicates how far the function reaches above and below its midline. For the function f(θ) = 2sin(2θ), the amplitude is 2, meaning the function oscillates between -2 and 2.

Period

The period of a function is the length of one complete cycle of the wave. For sine and cosine functions, the period can be determined from the coefficient of the variable inside the function. In f(θ) = 2sin(2θ), the period is calculated as 2π divided by the coefficient of θ, which is 2, resulting in a period of π.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe relationships between angles and sides of triangles. They are periodic functions, meaning they repeat their values in regular intervals. Understanding their properties, including amplitude and period, is essential for analyzing wave-like behaviors in various applications.

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