Solve the following equation on the interval [0,2pi) if sin = -0.29?
you know that sin(x) is negative in QIII and QIV
sin(.294) = 0.29
so, you want the numbers
π+.294 and 2π-.294
To solve the equation sin(x) = -0.29, we need to find the values of x on the interval [0, 2π) that satisfy this equation.
To do this, we can use the inverse sine function, also known as arcsin or sin^(-1), to find the angles whose sine is equal to -0.29.
The inverse sine function takes a value between -1 and 1 and returns the angle (in radians) whose sine is equal to that value.
So, we can write:
x = arcsin(-0.29)
To calculate the value of arcsin(-0.29), we can use a scientific calculator or a math software program that has the arcsine function. Let me calculate it for you.
Using a calculator or software, the arcsin(-0.29) is approximately -0.2867 radians.
Since we want to find the values of x on the interval [0, 2π), we need to add a multiple of 2π to the solution.
The general solution to the equation sin(x) = -0.29 is:
x = -0.2867 + 2πn, where n is an integer.
To find the specific values of x on the interval [0, 2π), we can substitute different values of n and calculate the corresponding x.
Let's substitute n = 0, 1, 2, ... and find the corresponding values of x:
For n = 0: x = -0.2867 + 2π(0) = -0.2867
For n = 1: x = -0.2867 + 2π(1) ≈ 5.995
For n = 2: x = -0.2867 + 2π(2) ≈ 12.277
...
And so on.
We continue substituting different values of n until we find all the values of x on the interval [0, 2π) that satisfy the equation sin(x) = -0.29.
Therefore, the solution to the equation sin(x) = -0.29 on the interval [0, 2π) is x ≈ -0.2867, 5.995, 12.277, ... and so on.