Solve on the interval $[0,2 \pi)$.
$4 \cos ^2 x-1=0$
A. $\left{\frac{\pi}{3}, \frac{2 \pi}{3}, \frac{4 \pi}{3}, \frac{5 \pi}{3}\right}$
B. $\left{\frac{\pi}{3}, \frac{2 \pi}{3}\right}$
C. $\left{\frac{\pi}{3}, \frac{5 \pi}{3}\right}$
D. There are no solutions in the given interval.
E. $\left{\frac{\pi}{6}, \frac{11 \pi}{6}\right}$
F. $\left{\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{7 \pi}{6}, \frac{11 \pi}{6}\right}$
Answer (2)
Isolate cos 2 x : cos 2 x = 4 1 .
Take the square root: cos x = ± 2 1 .
Solve for cos x = 2 1 : x = 3 π , 3 5 π .
Solve for cos x = − 2 1 : x = 3 2 π , 3 4 π . The solution is { 3 π , 3 2 π , 3 4 π , 3 5 π } .
Explanation
Understanding the Problem We are asked to solve the trigonometric equation 4 cos 2 x − 1 = 0 on the interval [ 0 , 2 π ) . This means we need to find all values of x between 0 and 2 π (inclusive of 0 , but exclusive of 2 π ) that satisfy the given equation.
Isolating cos 2 x First, let's isolate the cos 2 x term. We have:
4 cos 2 x − 1 = 0
Add 1 to both sides:
4 cos 2 x = 1
Divide by 4:
cos 2 x = 4 1
Taking the Square Root Now, take the square root of both sides of the equation:
cos 2 x = 4 1
This gives us:
cos x = ± 2 1
Two Cases to Consider We now have two separate equations to solve:
cos x = 2 1
cos x = − 2 1
Solving cos x = 2 1 For cos x = 2 1 , we know that cosine is positive in the first and fourth quadrants. The reference angle for x is 3 π . Therefore, the solutions in the interval [ 0 , 2 π ) are:
x = 3 π (first quadrant)
x = 2 π − 3 π = 3 6 π − 3 π = 3 5 π (fourth quadrant)
Solving cos x = − 2 1 For cos x = − 2 1 , we know that cosine is negative in the second and third quadrants. The reference angle for x is 3 π . Therefore, the solutions in the interval [ 0 , 2 π ) are:
x = π − 3 π = 3 3 π − 3 π = 3 2 π (second quadrant)
x = π + 3 π = 3 3 π + 3 π = 3 4 π (third quadrant)
Combining the Solutions Combining all solutions, we have:
x = 3 π , 3 2 π , 3 4 π , 3 5 π
Final Answer Therefore, the solution set is { 3 π , 3 2 π , 3 4 π , 3 5 π } .
Examples
Trigonometric equations are used in physics to model oscillatory motion, such as the motion of a pendulum or the vibration of a string. They are also used in engineering to design and analyze electrical circuits and mechanical systems. For example, the equation 4 cos 2 x − 1 = 0 could represent a simplified model of the voltage in an AC circuit, and solving for x would give the times at which the voltage reaches a certain level.
The solutions to the equation 4 cos 2 x − 1 = 0 on the interval [ 0 , 2 π ) are {\frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}}. Therefore, the correct answer is option A.
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