How do you solve 6x + 12 + 2cosx = 0?
6x +12 + 2cosx = 0
or
2cosx = -6x - 12
No straight-forward or easy way to get a reasonably accurate number.
Let's start by looking at the graphs of
y = 2cosx and y = -6x-12
http://www.wolframalpha.com/input/?i=plot+y+%3D+2cosx+,+y+%3D+-6x-12
looks like there is a solution near -2
how about Newton's Method:
dy/dx = 6 - 2sinx
newx = x - (6x+12+2cosx)/(6-2sinx)
where x is some starting value
begin with x = -2 , set your calculator to radians
newx ---- x
-1.893549... -1.894099..
-1.894099.. -1.8940998...
not bad after 2 steps.
x = appr -1.8941
verification:
http://www.wolframalpha.com/input/?i=sove+6x+%2B+12+%2B+2cosx+%3D+0
Newton's method converges rapidly. If you don't have calculus, then there are other methods, such a bisection.
f(x) = 6x + 12 + 2cosx
Start off with the interval (-3,-1) and find f at the midpoint: -2
f(-2) = 6(-2)+12+2cos(-2) = -0.832
So, f=0 occurs in (-2,-1).
f(-1.5) = 3.141
oops, too far, so f=0 in (-2,-1.5)
f(-1.75) = 1.14
f=0 in (-2,-1.75)
f(-1.875) = 0.151
Getting closer, but you can see that it will take a lot longer than Newton.
To solve the equation 6x + 12 + 2cosx = 0, you can follow these steps:
Step 1: Move all the terms containing x to one side of the equation:
6x + 2cosx = -12
Step 2: Divide both sides of the equation by 2 to isolate the cosx term:
3x + cosx = -6
Step 3: Rearrange the equation to solve for x:
cosx = -3x - 6
Step 4: Since it can be difficult to find the exact solutions for a trigonometric equation, we can use a graphing calculator or software to approximate the values of x. Alternatively, you can use numerical methods such as the Newton-Raphson method to find the approximate solutions.
Please note that the solutions to this equation may not be easily expressible in terms of simple radicals or known constants.
To solve the equation 6x + 12 + 2cos(x) = 0, we can use a combination of algebraic manipulation and graphical analysis. Here's how to approach it:
1. Start by subtracting 12 from both sides of the equation to isolate the terms involving "x":
6x + 2cos(x) = -12.
2. Since the equation involves cosine, which is a periodic function, we need to look for the values of x that satisfy the equation within a specific interval. Let's consider the interval [0, 2π] to find all the solutions in this range.
3. Now, we will utilize graphical analysis to estimate the solutions. Plot both sides of the equation on a graph. On the left-hand side, plot the line y = 6x, and on the right-hand side, plot y = -2cos(x) - 12.
4. The solutions to the equation correspond to the x-values where the graphs intersect. By observing the graph, we can see approximately where the intersecting points lie.
5. To obtain a more precise answer, use numerical methods like the Newton-Raphson method or the bisection method. These techniques can give you a numerical approximation of the solutions by iteratively refining your guesses.
By following these steps, you can solve the equation 6x + 12 + 2cos(x) = 0. Remember to use graphical analysis to estimate the solutions and then proceed with numerical methods for more accurate results.