The height h in inches of pistons 1 and 2 in an automobile engine can be modeled by h1 = 3.75sin(733t) + 7.5 and h2 = 3.75sin[733(t+ 4π/3)] + 7.5 where t is measured in seconds. How often are these two pistons at the same height? Show work please.

you would want:

3.75sin(733t) + 7.5 = 3.75sin[733(t+ 4π/3)] + 7.5
divide each term by 3.75, (how convenient ?)

sin(733t) + 2 = sin[733(t+ 4π/3)] + 2
sin(733t) = sin[733(t+ 4π/3)]
this can only be true if

733t = 733(t + 4π/3) OR 733(t+4π/3) = π - 733t
divide by 733
t = t + 4π/3 , not possible!

or
733(t+4π/3) = π - 733t
733t + 2932π/3 = π - 733t
1466t = -2929/3 π
t = appr -2.092 ---> this works in the original equation

period of each curve = 2π/733 = appr .00857

To find when the two pistons are at the same height, we need to set the two height equations equal to each other and solve for t.

Setting h1 = h2, we have:

3.75sin(733t) + 7.5 = 3.75sin[733(t+ 4π/3)] + 7.5

First, let's simplify the equation by subtracting 7.5 from both sides:

3.75sin(733t) = 3.75sin[733(t+ 4π/3)]

Next, let's divide both sides by 3.75 to isolate the sine function:

sin(733t) = sin[733(t+ 4π/3)]

Now, we can use the trigonometric identity sin(x) = sin(π - x) to simplify the equation further. This identity states that the sine of an angle is equal to the sine of the supplement of that angle.

sin(733t) = sin[733(t+ 4π/3)]
sin(733t) = sin[733t + 4π - 4π/3]
sin(733t) = sin[733t - 4π/3]

Since sin(x) = sin(y) implies x = 2πn + y or x = (2n + 1)π - y, where n is an integer, we can set the two arguments of sin equal to each other:

733t = 733t - 4π/3

Now, let's solve for t:

733t - 733t = -4π/3

0 = -4π/3

This equation has no solutions since -4π/3 is not equal to zero.

Therefore, the two pistons never coincide or are at the same height.

To find out how often the two pistons are at the same height, we need to solve the equation:

h1 = h2

Substituting the given expressions for h1 and h2:

3.75sin(733t) + 7.5 = 3.75sin[733(t+ 4π/3)] + 7.5

We can simplify the equation by canceling out the common terms (7.5) from both sides:

3.75sin(733t) = 3.75sin[733(t+ 4π/3)]

Now, divide both sides of the equation by 3.75:

sin(733t) = sin[733(t+ 4π/3)]

We have a trigonometric equation with sin on both sides. To solve this, we can use the sine function property that states:

sin(a) = sin(b) if and only if a = b + 2πn or a = π - b + 2πn

Where n is an integer.

Applying this property to our equation, we have two cases:

Case 1: 733t = 733(t+ 4π/3) + 2πn
Case 2: 733t = π - 733(t+ 4π/3) + 2πn

Now, let's solve each case individually:

Case 1:
733t = 733t + 977π/3 + 2πn

Simplifying the equation, we get:
0 = 977π/3 + 2πn

Divide both sides by π to isolate the n:
0 = (977/3) + 2n

Rearranging the equation, we find the value of n:
n = -977/6

But n must be an integer, so there is no solution for this case.

Case 2:
733t = π - 733t - 977π/3 + 2πn

Simplifying it further:
2 * 733t = π - 977π/3 + 2πn

7894t = π - 977π/3 + 2πn

To express the equation in terms of t:
t = (π - 977π/3 + 2πn) / 7894

Now, we need to find the values of t that satisfy this equation. We can plug in different integer values for n and solve for t.

Keep in mind that the time period of a sine function is 2π/733. So, to find the time when the two pistons are at the same height, we need to find the values of t within one period.

By plugging in different values for n between -733/2 and 733/2, we can calculate the corresponding values of t. For each valid value of t, we can find the corresponding time when the pistons are at the same height.

For example, let's try n = 0:

t = (π - 977π/3 + 2π(0)) / 7894
t = π - 977π/3 / 7894

Calculating this expression will give you a specific value for t. Repeat this process for different values of n within the given range to find all the times when the two pistons are at the same height.

Note: Due to the complexity of the equation, solving for exact solutions might not be feasible. A numerical approach or using a computer program can help find and approximate the solution.