Can someone help me solve this problem:
cos^2x + 5cosx - 2= 0
I don't know how to find x
Sure, I can help you solve this problem. To find the value of x in the equation cos^2x + 5cosx - 2 = 0, we can use a technique called factoring. Here's how you can do it:
Step 1: Rewrite the equation in the form of quadratic equation by replacing cos^2x with (cosx)^2. The equation becomes (cosx)^2 + 5cosx - 2 = 0.
Step 2: Now, let's try to factorize the quadratic equation. Look for two numbers that add up to 5 (the coefficient of cosx) and multiply to give -2 (the constant term). In this case, the numbers are -1 and 2.
Step 3: Rewrite the quadratic equation using the factored form. You can replace 5cosx with -1cosx + 2cosx. The equation becomes (cosx)^2 - cosx + 2cosx - 2 = 0.
Step 4: Group the terms in pairs and factor by grouping. The equation can be rearranged as [ (cosx)^2 - cosx ] + [ 2cosx - 2 ] = 0.
Step 5: Now, factor out common terms from each pair. (cosx)^2 - cosx can be factored as cosx(cosx - 1), and 2cosx - 2 can be factored as 2(cosx - 1). The equation becomes cosx(cosx - 1) + 2(cosx - 1) = 0.
Step 6: Notice that (cosx - 1) is a common factor in both terms. Factor it out: (cosx - 1)(cosx + 2) = 0.
Step 7: Equate each factor to zero.
cosx - 1 = 0
cosx + 2 = 0
Step 8: Solve for x in each equation.
cosx = 1
cosx = -2
Step 9: Since the range of cosine values lies between -1 and 1, the equation cosx = -2 has no solutions. However, cosx = 1 at x = 0.
Therefore, the solution to the equation cos^2x + 5cosx - 2 = 0 is x = 0.