solve the equation 4cos^2x-1=0
cos^2x = 1/4
cosx = ± 1/2
do you know an angle x , so that cosx = 1/2 ???
and by the CAST rule, isn't that angle in any of the 4 quadrants?
hint: cos60° = 1/2
To solve the equation 4cos^2x - 1 = 0, we can follow these steps:
Step 1: Rearrange the equation.
Start by adding 1 to both sides to isolate the term 4cos^2x.
This gives us: 4cos^2x = 1.
Step 2: Divide by 4.
Divide both sides of the equation by 4 to isolate the cosine term.
This gives us: cos^2x = 1/4.
Step 3: Take the square root.
To remove the square from the cosine term, take the square root of both sides of the equation.
This gives us: cosx = ±√(1/4).
Step 4: Simplify and solve.
The square root of 1/4 is 1/2, so we have two cases to consider:
1) cosx = 1/2
2) cosx = -1/2
To find the solutions for x, we need to find the angles that have a cosine value of 1/2 and -1/2. We can use the unit circle or trigonometric identities to identify these angles.
For case 1, cosx = 1/2:
- The angle whose cosine is 1/2 is 60 degrees (or π/3 radians) in the positive direction.
- The angle whose cosine is also 1/2 is 300 degrees (or 5π/3 radians) in the negative direction.
For case 2, cosx = -1/2:
- The angle whose cosine is -1/2 is 120 degrees (or 2π/3 radians) in the positive direction.
- The angle whose cosine is also -1/2 is 240 degrees (or 4π/3 radians) in the negative direction.
Therefore, the solutions for x are:
x = 60 degrees + 360n degrees or x = π/3 + 2πn, where n is an integer.
x = 300 degrees + 360n degrees or x = 5π/3 + 2πn, where n is an integer.
x = 120 degrees + 360n degrees or x = 2π/3 + 2πn, where n is an integer.
x = 240 degrees + 360n degrees or x = 4π/3 + 2πn, where n is an integer.
These solutions represent the multiple values of x that satisfy the equation 4cos^2x - 1 = 0.