Finding all Solutions for
2cos(x)-�ã3=0
Is this cosx=1/2sqrt3 ?
no it's 2cosx-sqrt3=0
I will read it as
2cosx - √3 = 0
cosx = √3/2
x = 30° , from the standard 30-60-90 triangle
or x = 330°
"all solutions " would be
x = 30 + k(360) or x = 330+k(360)° where k is an integer.
How do you find the solutions?
since the standard cosine curve has a period of 360°, any answer you have will repeat itself in the next curve or 360° either to the right or to the left.
by multiplying 360 by k, k and integer, I am simply adding/or subtracting multiples of 360 to any answer.
to get my original 30°, as I said, I recognized the rations of the 30-60-90 right-angled triangle, which are 1:√3:2.
Secondly , since the cosine is positive in the 1st and 4th quadrant, the 30° will be the 1st quadrant solution and 360-30 or 330° would be the 4th quadrant solution.
To find all the solutions for the equation 2cos(x) - √3 = 0, you need to solve for x. Let's break it down step by step:
Step 1: Add √3 to both sides of the equation:
2cos(x) = √3
Step 2: Divide both sides of the equation by 2:
cos(x) = √3 / 2
Now we need to find the values of x that satisfy this equation. In order to do that, we will use the inverse cosine function (also known as arccos or cos^(-1)). The inverse cosine function gives us the angle whose cosine equals a given value.
If we take the inverse cosine of both sides of the equation, we get:
x = arccos(√3 / 2)
Now, the inverse cosine function has a principal range of [0, π], which means it gives us angles between 0 and π (180 degrees). However, keep in mind that the cosine function is periodic with a period of 2π (360 degrees).
To find all the solutions within one full period, we need to account for both positive and negative values of x. Therefore, we have two solutions:
Solution 1: x = arccos(√3 / 2)
Solution 2: x = -arccos(√3 / 2)
The exact values of the angles can be found using the unit circle or a calculator. By plugging in √3 / 2 into the inverse cosine function, you would get:
Solution 1: x = π/6 (30 degrees)
Solution 2: x = -π/6 (-30 degrees)
After finding these solutions, you can continue to find other solutions by adding or subtracting multiples of the period (2π, or 360 degrees) from these initial solutions.
For example:
Solution 1: x = π/6 (30 degrees)
Adding 2π to the solution:
x = π/6 + 2π = 13π/6 (390 degrees)
Similarly, you can find the other solutions by adding or subtracting multiples of 2π to the initial solutions.
So, the general solution for the equation 2cos(x) - √3 = 0 is:
x = π/6 + 2nπ and x = -π/6 + 2nπ,
where n is an integer, representing any integer value for additional full periods.