Find all solutions of the equation in the interval [0,2pi).
sec(theta)+2=0
Write your answer in radians in terms of pi
sec x = -2, so
cos x = -1/2
solutions in QII and QIII
how do I write this in radians in radians in terms of pi
Well, well, well, let's tackle this equation and find some solutions that will make you smile! So we have sec(theta) + 2 = 0. To find the solutions in the interval [0, 2pi), we need to unwrap this equation.
First, we'll subtract 2 from both sides to isolate sec(theta), giving us sec(theta) = -2. Now, remember that sec(theta) is the reciprocal of cos(theta). So if sec(theta) equals -2, that means cos(theta) = -1/2.
Now, who's hanging out with cos(theta) at -1/2 in the interval [0, 2pi)? That's right, it's our good old friends pi/3 and 5pi/3! These are the solutions that will make our equation burst into laughter.
So, in radians, the solutions are theta = pi/3 and theta = 5pi/3.
Keep smiling and enjoy those solutions!
To find all solutions of the equation sec(theta) + 2 = 0 in the interval [0, 2pi), we'll follow these steps:
Step 1: Rewrite the equation in terms of cosine.
Since sec(theta) is the reciprocal of cosine(theta), we can rewrite the equation as follows:
1/cos(theta) + 2 = 0
Step 2: Combine the terms on the left side.
Multiply both sides of the equation by cos(theta) to eliminate the fraction:
1 + 2cos(theta) = 0
Step 3: Rearrange the equation.
Subtract 1 from both sides of the equation:
2cos(theta) = -1
Step 4: Solve for cos(theta).
Divide both sides of the equation by 2:
cos(theta) = -1/2
Step 5: Find the related angle in the unit circle.
The value -1/2 intersects the unit circle at two points: π/3 and 5π/3, where cos(theta) = -1/2.
Step 6: Determine the solutions in the given interval.
Since the interval is [0, 2pi), we need to find the values of theta that fall within this interval. The two solutions we found (π/3 and 5π/3) both fall within this interval.
Therefore, the solutions to the equation sec(theta) + 2 = 0 in the interval [0, 2pi) are:
θ = π/3 and θ = 5π/3, written in radians in terms of pi.
c'mon, guy.
cos π/3 = 1/2, so
cos 2π/3 = cos 4π/3 = -1/2
x = 2π/3 or 4π/3