Find all solutions of the equation in the interval [0,2pi) .
cos(theta)=0.7125
If there is more than one solution, separate them with commas.
Do not round any intermediate computations, and round your answer(s) to the nearest hundredth.
Can someone help me with this? Thanks
my first thought is what unit for angles? degrees, radians? I have never liked computer instruction, they demand precision to the hundreth place, but don't specify the units. It makes a difference.
Cosine is positive in first, fourth quadrant.
arccos.7125=44.56 degrees
so in the fourth quadrant, it is
360-44.56 degrees.
To find all solutions of the equation cos(theta) = 0.7125, we need to use the inverse cosine function (also known as arccosine). The inverse cosine function takes the cosine value as input and gives the angle as output.
Step 1: Take the inverse cosine of both sides of the equation:
arccos(cos(theta)) = arccos(0.7125)
Step 2: Simplify the left side using the inverse and direct cosine relation:
theta = arccos(0.7125)
Step 3: Use a calculator to find the inverse cosine of 0.7125. Make sure your calculator is in radian mode. The answer is approximately 0.7606.
Step 4: Since the given interval is [0,2pi), we need to check if there are any other solutions within this interval. We can add 2π (a full revolution) to the previous solution and check if it falls within the interval.
theta2 = theta + 2π = 0.7606 + 2π = 6.0032
Step 5: Round the solutions to the nearest hundredth:
The solutions in the interval [0,2π) are approximately 0.76 and 6.00.
Therefore, the solutions of the equation cos(theta) = 0.7125 in the interval [0,2π) are 0.76 and 6.00.
To find all the solutions of the equation cos(theta) = 0.7125 in the interval [0, 2pi), you can follow these steps:
1. Start by finding the principal solution of the equation. Take the inverse cosine (also known as arccosine) of both sides of the equation: theta = arccos(0.7125).
Note: The arccos function gives the angle whose cosine is the given value.
2. Use a calculator to find the arccos(0.7125). The result should be approximately 0.7671 radians.
3. Now, you have found the principal solution, which is theta = 0.7671 radians. However, in the interval [0, 2pi), there may be additional solutions.
4. To find the additional solutions, you need to consider the periodic nature of the cosine function. The cosine function repeats itself every 2pi radians. So, you can add or subtract multiples of 2pi to the principal solution and get equivalent solutions.
5. To find these additional solutions, add or subtract 2pi from the principal solution: theta = 0.7671 + 2pi*n or theta = 0.7671 - 2pi*n, where n is an integer.
6. To determine the possible values of n, you need to consider the interval [0, 2pi). Start by dividing the interval by 2pi: [0, 2pi) / 2pi = [0, 1).
7. Since n should be an integer, you can use the values n = 0 and n = 1, which fall within the interval [0, 1). So, you have two corresponding solutions:
Solution 1: theta = 0.7671 + 2pi * 0 = 0.7671 radians (the principal solution)
Solution 2: theta = 0.7671 + 2pi * 1 = 6.0097 radians (after rounding to the nearest hundredth)
8. Therefore, the solutions to the equation cos(theta) = 0.7125 in the interval [0, 2pi) are approximately 0.77 and 6.01 radians.
Remember to round your answers to the nearest hundredth, as requested in the question.