solve an equation in the interval from 0 to 2π. Round to the nearest hundredth. 4cos(t)=3
no words just the steps
sorry, but
cos(t) = 3/4
has only two solutions, in QI and QIV
You are correct. I apologize for the mistake in my previous response.
Here are the corrected steps:
1. Start with the equation: 4cos(t) = 3
2. Divide both sides by 4: cos(t) = 3/4
3. Take the inverse cosine of both sides: t = ±cos⁻¹(3/4) + 2πk
4. Convert cos⁻¹(3/4) to a decimal using a calculator: cos⁻¹(3/4) ≈ 0.72
5. Since cosine is positive in Quadrant I and Quadrant IV, we only need to consider those two quadrants.
6. Plug in values of k to get solutions in the interval from 0 to 2π:
- for t ≈ 0.72 + 2πk: k = 0. t ≈ 0.72
- for t ≈ -0.72 + 2πk: k = 0, t ≈ 2π - 0.72
7. Round the solutions to the nearest hundredth: t ≈ 0.72, 5.56
8. Therefore, the solutions in the interval from 0 to 2π are t ≈ 0.72 and t ≈ 5.56.
To solve the equation 4cos(t) = 3 in the interval from 0 to 2π, we can isolate the cosine function by dividing both sides of the equation by 4:
(4cos(t))/4 = 3/4
cos(t) = 3/4
To find the value of t that satisfies this equation, we need to take the inverse cosine (also known as arccos) of both sides:
t = arccos(3/4)
Using a calculator, we can find the value of arccos(3/4) to be approximately 0.7227 radians.
However, we need to round this value to the nearest hundredth within the interval from 0 to 2π.
Since 2π is approximately 6.2832, we need to check if the value 0.7227 is within this interval.
0 ≤ 0.7227 ≤ 6.2832
Since the value 0.7227 is within the interval, the solution to the equation 4cos(t) = 3 in the interval from 0 to 2π rounded to the nearest hundredth is t ≈ 0.72.
To solve the equation 4cos(t) = 3 on the interval from 0 to 2π, we can follow these steps:
1. Divide both sides of the equation by 4 to isolate the cosine function: cos(t) = 3/4.
This step helps us find the values of t where the cosine function equals 3/4.
2. Take the inverse cosine (arccos) of both sides of the equation: t = arccos(3/4).
This step helps us find the angles (t) which have a cosine value of 3/4.
Now, let's determine the values of t in the given interval using a calculator:
- On your calculator, find the inverse cosine function. It may be labeled as "cos^(-1)" or "arccos".
- Input 3/4 as the argument for the inverse cosine function: arccos(3/4).
The calculator will give you the value of arccos(3/4) in radians. It will likely be a decimal number. Now, we need to find all the values of t in the interval from 0 to 2π, rounded to the nearest hundredth.
Since cos function is periodic with a period of 2π, we can find all the solutions by adding or subtracting multiples of 2π from the initial solution we obtained from the calculator.
For example, if the calculator gives us t = 0.725, we can find all the solutions in the interval from 0 to 2π by adding or subtracting multiples of 2π:
t = 0.725 + 2πk and t = -0.725 + 2πk, where k is an integer.
By repeating this process for all the values you obtained from the calculator, you will find the solutions of the equation 4cos(t) = 3 in the interval from 0 to 2π, rounded to the nearest hundredth.
Starting with 4cos(t) = 3:
cos(t) = 3/4
Taking the inverse cosine of both sides:
t = ±cos⁻¹(3/4) + 2πk, where k is an integer.
Using a calculator, we find that cos⁻¹(3/4) ≈ 0.72.
Therefore, t ≈ 0.72 + 2πk or t ≈ -0.72 + 2πk for some integer k.
Since we are looking for solutions in the interval from 0 to 2π, we can use these two equations to find the values of k that give solutions in that interval.
For t ≈ 0.72 + 2πk, we need k to be 0, 1, or -1 in order to get a solution in the desired interval. Plugging in those values of k, we get approximately:
t ≈ 0.72
t ≈ 6.05
t ≈ 3.46
For t ≈ -0.72 + 2πk, we need k to be 0, 1, or -1 in order to get a solution in the desired interval. Plugging in those values of k, we get approximately:
t ≈ 5.41
t ≈ 2.83
t ≈ 0.25
Rounding all of these answers to the nearest hundredth, we get:
t ≈ 0.72, 0.25, 2.83, 3.46, 5.41, 6.05
Therefore, the solutions in the interval from 0 to 2π are t ≈ 0.72, 0.25, 2.83, 3.46, 5.41, and 6.05.
1. Start with the equation: 4cos(t) = 3
2. Divide both sides by 4: cos(t) = 3/4
3. Take the inverse cosine of both sides: t = ±cos⁻¹(3/4) + 2πk
4. Convert cos⁻¹(3/4) to a decimal using a calculator: cos⁻¹(3/4) ≈ 0.72
5. Plug in values of k to get solutions in the interval from 0 to 2π:
- for t ≈ 0.72 + 2πk: k = 0, 1, -1. t ≈ 0.72, 6.05, 3.46
- for t ≈ -0.72 + 2πk: k = 0, 1, -1. t ≈ 5.41, 2.83, 0.25
6. Round the solutions to the nearest hundredth: t ≈ 0.72, 0.25, 2.83, 3.46, 5.41, 6.05
7. Therefore, the solutions in the interval from 0 to 2π are t ≈ 0.72, 0.25, 2.83, 3.46, 5.41, and 6.05.