y=cos(square root of 4t+12)
dy/dt=-sin(square root of 4t+12)*d/dx(sqare root of 4t+12)
=-sin(sqare root of 4t+12)*1/2*sqare root of 4t+12 * 4
=-sin (sqare root of 4t+12)* 2/sqare root of 4t+12
To simplify the given equation, we can take it step by step. Let's start with the given equation:
y = cos(√(4t + 12))
Step 1: Simplify the square root expression inside the cosine function.
√(4t + 12) = √4 * √(t + 3)
= 2√(t + 3)
Now, we have:
y = cos(2√(t + 3))
Step 2: The cosine function repeats every 2π radians, so we can convert the given expression using the period of cosine.
The period of the cosine function is 2π/b, where b is the coefficient of the angle inside the function.
In this case, b = 2, so the period is:
T = 2π/2
= π
Step 3: Rewrite the given equation in terms of the new period.
y = cos(2√(t + 3))
= cos(2 * (√(t + 3)))
= cos(2 * (√(t + 3) * (√(t + 3))/√(t + 3))
= cos(2 * (√(t + 3)^2)/√(t + 3))
= cos(2(t + 3)/√(t + 3))
= cos(2(t + 3)/2√(t + 3))
= cos((t + 3)/√(t + 3))
Now, the simplified equation is:
y = cos((t + 3)/√(t + 3))
This is the simplified form of the given equation.
The equation you provided is y = cos(√(4t + 12)). This is a mathematical expression involving the cosine function. To evaluate this equation, you need to substitute a value for t.
For example, if you want to find the value of y when t = 2, you can substitute t = 2 into the equation:
y = cos(√(4 * 2 + 12))
y = cos(√(8 + 12))
y = cos(√(20))
y = cos(√20)
Now, to find the numerical value of y, you can use a calculator or software with a cosine function. Here are the steps to evaluate the cosine function with square root:
1. Calculate the square root of the argument: √20 ≈ 4.47.
2. Use the cosine function with the calculated value: cos(4.47) ≈ -0.2108.
So, when t = 2, the value of y is approximately -0.2108.
You can repeat this process for different values of t to get corresponding values of y.