Determine approximate solutions for this equation in the interval x is all real numbers [0, 2pi), to the nearest hundredth of a radian:
cosx + 0.75 = 0
cosx = -.75
the cosine is negative in II and III
take inverse cosine of +.75, this will give you the reference angle
for angle in II, take pi - reference angle
for angle in III take pi + reference angle
To find approximate solutions for the equation cos(x) + 0.75 = 0 in the interval [0, 2π), we can follow these steps:
1. Subtract 0.75 from both sides of the equation:
cos(x) = -0.75
2. Take the inverse cosine (arccos) of both sides:
x = arccos(-0.75)
3. To find the principal value of x, evaluate arccos(-0.75) using a calculator:
x ≈ 2.4981 radians
4. Since the interval is [0, 2π), we need to find any additional solutions within this range. To do this, subtract the principal value from 2π:
2π - 2.4981 ≈ 3.7845 radians
5. Therefore, the approximate solutions for the equation in the given interval are:
x ≈ 2.4981 radians and x ≈ 3.7845 radians (to the nearest hundredth of a radian).
To solve the equation cos(x) + 0.75 = 0, we need to find the values of x where the cosine of x, added to 0.75, equals zero.
Step 1: Subtract 0.75 from both sides of the equation to isolate the cosine term:
cos(x) = -0.75
Step 2: Take the inverse cosine (arccos) of both sides to find the angle x:
x = arccos(-0.75)
Step 3: Use a calculator to find the arccosine of -0.75. Most calculators have a built-in function for finding the inverse cosine. Enter -0.75 into your calculator, then press the arccos or cos⁻¹ button. Let's denote the result as A:
A = arccos(-0.75)
Step 4: Determine the possible solutions for x in the given interval [0, 2pi). The inverse cosine function returns a single value, but cosine has a periodic nature, repeating every 2π radians (or 360 degrees). So, we need to consider the full range of possible solutions within the given interval.
Step 5: Add or subtract 2π (360 degrees) multiples from the initial solution A to account for all the possible solutions within the interval.
Since we are only interested in the solutions to the nearest hundredth of a radian, substitute the value of A into the following formula to calculate the approximate solutions:
x1 = A (the original solution)
x2 = A + 2π
x3 = A - 2π
These three values, x1, x2, and x3, are the approximate solutions to the equation cos(x) + 0.75 = 0 in the interval [0, 2π) to the nearest hundredth of a radian.