Math  /  Trigonometry

QuestionWhich statements are true? Select all that apply. A. The amplitude of y=2sin(12x)y=2 \sin \left(\frac{1}{2} x\right) is 12\frac{1}{2} B. The period of y=2sin(12x)y=2 \sin \left(\frac{1}{2} x\right) is π\pi. C. The amplitude of y=2sin(12x)y=2 \sin \left(\frac{1}{2} x\right) is π2\frac{\pi}{2}. D. The period of y=2sin(12x)y=2 \sin \left(\frac{1}{2} x\right) is 4π4 \pi. E. The period of y=2sin(12x)y=2 \sin \left(\frac{1}{2} x\right) is 12\frac{1}{2} F. The amplitude of y=2sin(12x)y=2 \sin \left(\frac{1}{2} x\right) is 2

Studdy Solution

STEP 1

What is this asking? Which of these statements correctly describe the amplitude and period of the sine wave y=2sin(12x)y = 2 \sin(\frac{1}{2} x)? Watch out! Don't mix up amplitude and period!
Amplitude is how high the wave goes, and period is how long it takes to repeat.

STEP 2

1. Find the Amplitude
2. Find the Period

STEP 3

Alright, let's **decode** this sine wave!
Remember, the **standard form** of a sine function is y=Asin(Bx)y = A \sin(Bx), where A|A| is the **amplitude**.

STEP 4

In our equation, y=2sin(12x)y = 2 \sin(\frac{1}{2} x), the number multiplying the sine function is A=2A = \mathbf{2}.
So, the **amplitude** is 2=2|2| = \mathbf{2}.
That's how high and low the wave swings from its center!

STEP 5

Now, for the **period**!
The period of a sine function y=Asin(Bx)y = A \sin(Bx) is given by 2πB\frac{2\pi}{|B|}.
This tells us how long it takes for the wave to complete one full cycle.

STEP 6

In our equation, y=2sin(12x)y = 2 \sin(\frac{1}{2} x), we have B=12B = \frac{\mathbf{1}}{\mathbf{2}}.
Let's plug that into our period formula:
Period=2πB=2π12 \text{Period} = \frac{2\pi}{|B|} = \frac{2\pi}{|\frac{1}{2}|}

STEP 7

Since the absolute value of 12\frac{1}{2} is just 12\frac{1}{2}, we have:
Period=2π12 \text{Period} = \frac{2\pi}{\frac{1}{2}}

STEP 8

To divide by a fraction, we multiply by its reciprocal.
The reciprocal of 12\frac{1}{2} is 2\mathbf{2}, so:
Period=2π2=4π \text{Period} = 2\pi \cdot 2 = \mathbf{4\pi}

STEP 9

So, the **period** of our sine wave is 4π\mathbf{4\pi}!
That's the length of one complete wave cycle.

STEP 10

Statements **D** (4π4\pi is the period) and **F** (amplitude is 22) are **true**.
The rest are false!

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