Solve for exact or to 1d.p of x E [0 ,2pi]
4cos^2x-3
4(cosx)^2 - 3 = 0
Add 3 to each side
4(cosx)^2 = 3
Divide both sides by 4
(cosx)^2 = 3/4
Take the square root of each side
cosx = sqrt(3/4) or -sqrt(3/4)
cosx = sqrt(3)/2 or -sqrt(3)/2
x = pi/6, 11pi/6 or 5pi/6, 7pi/6
:)
To "solve" we need an equation.
I will assume you meant:
4cos^2 x - 3 = 0
cos^2 x = 3/4
cosx = ± √3/2 , ----> all 4 quadrants
you should recognize that cos π/6 = √3/2
x = π/6
or
x = π - π/6 = 5π/6
or
x = π + π/6 = 7π/6
or
x = 2π - π/6 = 11π/6
To solve for the exact or to 1 decimal place value of the expression 4cos^2(x) - 3 on the interval [0, 2π], we will follow these steps:
Step 1: Evaluate the expression for the given values of x within the interval [0, 2π].
Step 2: Round the result to either the exact value or 1 decimal place, as required.
Step 1: Evaluate the expression for x within the interval [0, 2π]:
For x = 0:
4cos^2(0) - 3 = 4(1) - 3 = 4 - 3 = 1
For x = π/2:
4cos^2(π/2) - 3 = 4(0^2) - 3 = 0 - 3 = -3
For x = π:
4cos^2(π) - 3 = 4(-1)^2 - 3 = 4(1) - 3 = 4 - 3 = 1
For x = 3π/2:
4cos^2(3π/2) - 3 = 4(0^2) - 3 = 0 - 3 = -3
For x = 2π:
4cos^2(2π) - 3 = 4(1)^2 - 3 = 4(1) - 3 = 4 - 3 = 1
Step 2: Round the result to either the exact value or 1 decimal place, as required:
Since we are solving for the exact or to 1 decimal place value, the rounded values are as follows:
For x = 0: 1
For x = π/2: -3
For x = π: 1
For x = 3π/2: -3
For x = 2π: 1
Therefore, the exact or rounded (to 1 decimal place) values of the expression 4cos^2(x) - 3 on the interval [0, 2π] are:
Exact values: {1, -3, 1, -3, 1}
Rounded values (to 1 decimal place): {1, -3, 1, -3, 1}