Solve for x:
5sin(2x) + cos(x) = -1.916
Can someone please show me the steps?
To solve the equation 5sin(2x) + cos(x) = -1.916, we need to isolate the variable x. Here are the steps:
Step 1: Rearrange the equation
Move the constant term (-1.916) to the other side of the equation:
5sin(2x) + cos(x) + 1.916 = 0
Step 2: Apply trigonometric identities
We can transform the equation using some trigonometric identities. First, let's rewrite the double-angle term using the identity sin(2x) = 2sin(x)cos(x):
5(2sin(x)cos(x)) + cos(x) + 1.916 = 0
Simplifying further:
10sin(x)cos(x) + cos(x) + 1.916 = 0
Step 3: Factor
Now we can factor out the common term cos(x):
cos(x)(10sin(x) + 1) + 1.916 = 0
Step 4: Solve for cos(x)
Since we want to solve for x, we need to isolate the cosine term. Subtract 1.916 from both sides of the equation:
cos(x)(10sin(x) + 1) = -1.916
Step 5: Solve for sin(x)
To solve for sin(x), we have two possibilities:
1. If cos(x) ≠ 0, then we divide both sides of the equation by cos(x):
10sin(x) + 1 = -1.916 / cos(x)
2. If cos(x) = 0, then sin(x) can be any real number. We will deal with this case separately.
Step 6: Solve for sin(x) (when cos(x) ≠ 0)
Now we solve the equation 10sin(x) + 1 = -1.916 / cos(x) for sin(x). Rearrange the equation:
10sin(x) = (-1.916 / cos(x)) - 1
Step 7: Divide by 10
Divide both sides of the equation by 10:
sin(x) = [(-1.916 / cos(x)) - 1] / 10
Step 8: Solve for x (when cos(x) ≠ 0)
To solve for x, we can apply the inverse sine function, also known as arcsin or sin^(-1), to both sides of the equation:
x = arcsin{[(-1.916 / cos(x)) - 1] / 10}
Step 9: Solve for x (when cos(x) = 0)
If cos(x) = 0, then sin(x) can be any real number. In this case, we need to solve the equation cos(x) = 0 separately.
Step 10: Solve for x (when cos(x) = 0)
To solve cos(x) = 0, we find the values of x where the cosine function equals zero. These values occur when x is equal to odd multiples of π/2:
x = (2n + 1)π/2, where n is an integer.
In conclusion, the general solution for x is given by x = arcsin{[(-1.916 / cos(x)) - 1] / 10} for cos(x) ≠ 0, and x = (2n + 1)π/2 for cos(x) = 0, where n is an integer.