Show your steps. Solve the equation in the interval from 0 to 2π. Round to the nearest hundredth. 7cos(2t) = 3
To solve for t in the interval from 0 to 2π we can use inverse cosine (cos⁻¹) and the following steps:
1. Divide both sides by 7: cos(2t) = 3/7
2. Take the inverse cosine of both sides: cos⁻¹(cos(2t)) = cos⁻¹(3/7)
3. Simplify: 2t = ±0.9344 + 2πn or 2t = ±2.207 + 2πn, where n is an integer
4. Solve for t: t = 0.4672 + πn or t = 1.1035 + πn, where n is an integer
Therefore, the solutions in the interval from 0 to 2π are:
t ≈ 0.47, 1.10, 2.68, 3.31, 4.89, and 5.53.
To solve the equation 7cos(2t) = 3 in the interval from 0 to 2π, we can follow these steps:
Step 1: Divide both sides of the equation by 7 to isolate the cosine function:
cos(2t) = 3/7
Step 2: Take the inverse cosine (or arccos) of both sides of the equation to eliminate the cosine function:
2t = arccos(3/7)
Step 3: Divide both sides of the equation by 2 to solve for t:
t = arccos(3/7) / 2
Step 4: Evaluate the arccos(3/7) / 2 value and round it to the nearest hundredth.
Using a calculator, arccos(3/7) ≈ 0.7754
Therefore, t ≈ 0.7754 / 2 ≈ 0.3877
So the solution to the equation 7cos(2t) = 3 in the interval from 0 to 2π is t ≈ 0.3877.