Question10. A given sinusoidal function has a period of 3 , an amplitude of 7 , and a maximum at , 2). Represent the function with a sine equation and a cosine equation. [4 marks]
Studdy Solution
STEP 1
1. The sinusoidal function has a period of 3.
2. The amplitude of the function is 7.
3. The function reaches its maximum value at the point .
4. We need to represent the function using both sine and cosine equations.
STEP 2
1. Determine the vertical shift of the function.
2. Determine the sine function representation.
3. Determine the cosine function representation.
STEP 3
The amplitude of the function is 7, which means the function oscillates 7 units above and below its midline. Since the maximum is at , the midline is:
The vertical shift is the midline value:
STEP 4
The general form of a sine function is:
Given:
- Amplitude
- Period , so
- Vertical shift
Since the sine function starts at the midline, we need to adjust it to reach the maximum at . Therefore, we need a phase shift such that:
This occurs when:
Thus, the sine equation is:
Simplified:
STEP 5
The general form of a cosine function is:
Given:
- Amplitude
- Period , so
- Vertical shift
Since the cosine function starts at the maximum, we can use it directly with no phase shift:
Thus, the cosine equation is:
The sine equation representing the function is:
The cosine equation representing the function is:
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