Solve the equation in the interval from 0 to 2π. Round to the nearest hundredth. 7cos(2t) = 3
We can start by isolating cos(2t) by dividing both sides by 7:
cos(2t) = 3/7
Next, we can use the inverse cosine function to solve for 2t:
2t = ±cos⁻¹(3/7)
Using a calculator, we find that cos⁻¹(3/7) is approximately 1.0201 radians. So:
2t = ±1.0201 + 2πn or 2t = ±(2π - 1.0201) + 2πn
where n is an integer. We can simplify this to:
t = ±0.5101 + πn or t = ±2.1200 + πn
Rounding to the nearest hundredth and plugging in n=0, 1, and 2, we get the following solutions in the interval from 0 to 2π:
t ≈ 0.51, 3.63 or t ≈ 2.12, 5.24
Therefore, the solutions to the equation 7cos(2t) = 3 in the interval from 0 to 2π are approximately 0.51, 2.12, 3.63, and 5.24.
To solve the equation 7cos(2t) = 3 in the interval from 0 to 2π, we need to isolate the variable t.
Step 1: Divide both sides of the equation by 7 to isolate the cosine term:
cos(2t) = 3/7
Step 2: Take the inverse cosine (arccos) of both sides:
2t = arccos(3/7)
Step 3: Divide both sides by 2 to solve for t:
t = (1/2) * arccos(3/7)
Step 4: To round to the nearest hundredth, evaluate the expression:
t ≈ (1/2) * arccos(3/7) ≈ 0.826
Therefore, the approximate solution to the equation 7cos(2t) = 3 in the interval from 0 to 2π, rounded to the nearest hundredth, is t ≈ 0.83.