The shape of an electromagnetic wave is modelled by the equation:
y=14cos(πx/57)+9
 
where  x
  is measured in radians.
a)
Write down the amplitude of this wave.
A=
 
14
interpreted as 14
 Your answer is correct
 

Your answer is numerically correct. You were awarded 1 mark.
You scored 1 mark for this part.
Score: 1/1Your answer is correctAnswered

b)
Calculate the period (wavelength) of this wave.
The period (wavelength) is 
114
interpreted as 114
 Your answer is correct
 .
Your answer is numerically correct. You were awarded 1 mark.
You scored 1 mark for this part.
Score: 1/1Your answer is correctAnswered

c)
Find the smallest value of  x
  for which  y=8
 .
x=
 
 
  radians.
Give your answer rounded to 2 decimal places, if needed.

To find the smallest value of \( x \) for which \( y = 8 \) in the equation \( y = 14\cos\left(\frac{\pi x}{57}\right) + 9 \), we need to solve the equation:

\[
8 = 14\cos\left(\frac{\pi x}{57}\right) + 9
\]

1. Subtract 9 from both sides:

\[
8 - 9 = 14\cos\left(\frac{\pi x}{57}\right)
\]

\[
-1 = 14\cos\left(\frac{\pi x}{57}\right)
\]

2. Divide both sides by 14:

\[
\cos\left(\frac{\pi x}{57}\right) = -\frac{1}{14}
\]

3. Take the inverse cosine (arccos) of both sides:

\[
\frac{\pi x}{57} = \arccos\left(-\frac{1}{14}\right)
\]

4. Solve for \( x \):

\[
x = \frac{57}{\pi} \arccos\left(-\frac{1}{14}\right)
\]

5. Now, calculate this using a calculator:

\[
x \approx \frac{57}{\pi} \times \arccos\left(-\frac{1}{14}\right)
\]

Let's compute the value.

\[
\arccos\left(-\frac{1}{14}\right) \approx 1.643
\]

\[
x \approx \frac{57 \times 1.643}{\pi} \approx \frac{93.651}{3.1416} \approx 29.81 \text{ radians}
\]

So, the smallest value of \( x \) for which \( y = 8 \) is approximately \( x = 29.81 \) radians.

### Do you want any further details or have any questions?

Here are 5 related questions you might consider:
1. What is the general form of a cosine wave function and how does each parameter affect the wave?
2. How would the equation change if the wave were shifted vertically by 5 units?
3. Can you find the maximum value of \( y \) for this wave and the corresponding \( x \) value?
4. How would you solve for \( x \) if \( y = 20 \) instead of \( y = 8 \)?
5. How does changing the amplitude affect the period (wavelength) of the wave?

**Tip:** When solving trigonometric equations, always consider the range of the inverse trigonometric function to find the correct value of \( x \).

Explore how to find the amplitude, period, and smallest x value for the wave function y = 14 cos(πx/57) + 9. Understand the concepts of amplitude and wavelength in trigonometry with detailed step-by-step solutions.

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