Solve the multiple-angle equation. (Enter your answers as a comma-separated list. Let n be any integer. Enter your response in radians.)
sec 9x = 2
sec 9x = 2
cos 9x = 1/2
so 9x = π/3 or 5π/3
but period of cos 9x is 2π/9
so adding / subtracting any multiples of 2π/9 to an existing answer will yield a new answer
general solution:
x = π/3 + 2nπ/9 , 5π/3 + 2nπ/9
To solve the equation sec(9x) = 2, we need to find the values of x that satisfy the equation.
Step 1: Reciprocal Identity
Since sec(9x) = 2, we can rewrite it as 1/cos(9x) = 2.
Step 2: Finding the cosine value
To find the value of cos(9x), we take the reciprocal of 2, which is 1/2. So, cos(9x) = 1/2.
Step 3: Finding the angle
We are looking for angles where the cosine is equal to 1/2. The cosine of an angle of π/3 (60 degrees) is equal to 1/2.
Step 4: Multiple-angle formula
We can use the multiple-angle formula for cos(9x) to find the solutions. The formula is:
cos(9x) = 2cos^2(4.5x) - 1
Using this formula, we have:
1/2 = 2cos^2(4.5x) - 1
Simplifying the equation, we get:
2cos^2(4.5x) = 3/2
cos^2(4.5x) = 3/4
Taking the square root of both sides, we get:
cos(4.5x) = ±√(3/4)
cos(4.5x) = ±√3/2
Step 5: Finding the angle values
The cosine function is positive in the first and fourth quadrants. So, we can set up the following equations:
cos(4.5x) = √3/2
cos(4.5x) = -√3/2
Solving the first equation, we get:
4.5x = π/6 + 2πn, 11π/6 + 2πn
Solving the second equation, we get:
4.5x = 5π/6 + 2πn, 7π/6 + 2πn
where n is an integer.
Step 6: Solving for x
To find the values of x, we divide each equation by 4.5:
x = π/27 + 4πn/9, 11π/27 + 4πn/9
x = 5π/27 + 4πn/9, 7π/27 + 4πn/9
Therefore, the solutions for the equation sec(9x) = 2 are:
x = π/27 + 4πn/9, 11π/27 + 4πn/9, 5π/27 + 4πn/9, 7π/27 + 4πn/9
where n is an integer.
To solve the equation sec 9x = 2, we first need to isolate the angle 9x.
Step 1: Take the reciprocal of both sides to get cos 9x = 1/2.
Step 2: We know that the cosine function has a value of 1/2 at two different angles, π/3 and 5π/3 (as well as their coterminal angles).
To find the general solution for 9x, we need to solve the equation 9x = π/3 + 2πn and 9x = 5π/3 + 2πn, where n is any integer.
Step 3: Solve for x.
For 9x = π/3 + 2πn:
Divide both sides by 9: x = (π/3 + 2πn)/9
For 9x = 5π/3 + 2πn:
Divide both sides by 9: x = (5π/3 + 2πn)/9
Step 4: Simplify (if necessary).
The expressions (π/3 + 2πn)/9 and (5π/3 + 2πn)/9 can be left as they are since they are already in a simplified form.
Therefore, the general solution for x is x = (π/3 + 2πn)/9 and x = (5π/3 + 2πn)/9, where n is any integer.
Note: If you need specific values for x within a certain range, you can substitute different integer values for n into the equations to find the corresponding values of x within that range.