3x^2 - 2 + 2 sin(x) = 0


The easy way is to graph it, and look for the zeroes in the graph. My guess is there is one around 18 deg, and at about 55 deg.

I don't see offhand another way to solve it.

To solve the equation 3x^2 - 2 + 2sin(x) = 0, there are several methods you can use. Let's go through two common approaches: using a numerical method and using a graphical method.

1. Numerical Method (Approximate Solution):
One way to solve the equation numerically is by using an iterative method like the Newton-Raphson method or the Bisection method. These methods involve repeatedly estimating the root of the equation. Here's a brief explanation of the Bisection method:

a) First, rewrite the equation as f(x) = 3x^2 - 2 + 2sin(x) = 0.
b) Determine an interval [a, b] such that f(a) and f(b) have opposite signs. For example, let's use [0, 2] since f(0) is negative and f(2) is positive.
c) Divide the interval in half and evaluate the function at the midpoint, let's call it c. If f(c) = 0 or the function value is close to zero within a desired tolerance, consider c as the approximate root.
d) If f(a) and f(c) have opposite signs, set b = c. Otherwise, set a = c.
e) Repeat steps c and d until you obtain an approximate root within the desired tolerance.

2. Graphical Method:
To solve the equation graphically, you can plot the function f(x) = 3x^2 - 2 + 2sin(x) and check for the points where the graph intersects the x-axis (where f(x) = 0). This method gives you an estimate of the roots visually. Here are the steps:

a) Plot the function f(x) = 3x^2 - 2 + 2sin(x) on a graphing calculator or software.
b) Observe the points where the graph intersects or comes close to the x-axis (where f(x) = 0).
c) Estimate the x-values of these points and refine them if needed using the zoom feature of the graphing tool.

Both methods will help you approximate the solutions to the equation. Keep in mind that these are approximate solutions, and finding exact solutions may require more advanced techniques or mathematical analysis.