1) To solve the equation 2cos^2x = 1 + sin x, we can use the Pythagorean identity cos^2x = 1 - sin^2x Substitute cos^2x = 1 - sin^2x into the equation 2(1 - sin^2x) = 1 + sin Simplify 2 - 2sin^2x = 1 + sin Rearrange and rewrite in quadratic form 2sin^2x + sin x - 1 = Now, this is a quadratic equation in terms of sin x. Solve for sin x using the quadratic formula or by factoring.
2) To solve the equation 5^x + 5^(x+1) = 30, we first simplify the equation Rearrange the equation as 5^x + 55^x = 3 Combine like terms 65^x = 3 Divide both sides by 6 5^x = Now, we can see that x = 1 satisfies the equation. So, the solution to the equation is x = 1.
1) To solve the equation 2cos^2x = 1 + sin x, we can use the Pythagorean identity cos^2x = 1 - sin^2x
Substitute cos^2x = 1 - sin^2x into the equation
2(1 - sin^2x) = 1 + sin
Simplify
2 - 2sin^2x = 1 + sin
Rearrange and rewrite in quadratic form
2sin^2x + sin x - 1 =
Now, this is a quadratic equation in terms of sin x. Solve for sin x using the quadratic formula or by factoring.
2) To solve the equation 5^x + 5^(x+1) = 30, we first simplify the equation
Rearrange the equation as 5^x + 55^x = 3
Combine like terms
65^x = 3
Divide both sides by 6
5^x =
Now, we can see that x = 1 satisfies the equation. So, the solution to the equation is x = 1.