# When no vertical translation is considered, the sine curve can be represented by the function y = a sin (bx + c). If the area contained between the curve, the x axis and the ordinates at x = 0 and...

When no vertical translation is considered, the sine curve can be represented by the function y = a sin (bx + c).

If the area contained between the curve, the x axis and the ordinates at x = 0 and x = `pi` is 4 square units, investigate alternate values of a, b and c which would achieve this result.

Could you please help me solve this problem

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### 1 Answer

You want to find values for the parameters a,b, and c such that the area between the curve y=asin(bx+c) and the x-axis between 0 and pi is 4.

The area represented is the integral:

`A=int_0^pi asin(bx+c)dx ` where we take the signed area (area below the x-axis is treated as a negative.)

Then `int_0^pi asin(bx+c)dx=-a/b cos(bx+c)|_0^pi; b ne 0 `

Thus we want `-a/bcos(b pi + c) + a/b cos(c)=4 `

You would want to fix two of the parameters and solve for the third.

For example:

If a=2 c=0 then b=1 or 1/2. (Note that `c=2npi ` for natural number n)

If a=6, c=0 then b=3 or 3/2.

If a=18, c=0, b= 9 or 9/2.

If you start with a=9 and b=3/2 we can solve for c:

`-9/(3/2)(cos((3pi)/2)+c)+9/(3/2)cos(c)=4 `

`-6cos((3pi)/2+c)+6cos(c)=4 `

`cos((3pi)/2+c)-cos(c)=-2/3 ` Use the angle addition identity:

sin(c)-cos(c)=-2/3 Square both sides and use the Pythagorean identity:

`-2sin(c)cos(c)=-5/9 `

sin(2c)=5/18

2c is approximately .2814 so 1 result for c is .1407

Choose nice numbers and investigate what happens as you vary the parameters.

Note that a is a vertical dilation and b is a horizontal dilation while c is a vertical translation.

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