To solve the equation 3sin^2(x) + 7cos(x) - 3 = 0, we can use some trigonometric identities to simplify the expression.
Start by using the Pythagorean Identity: sin^2(x) + cos^2(x) = 1.
Rewrite the equation in terms of sin^2(x) only 3sin^2(x) = 3 - 3cos^2(x).
Substitute this into the original equation 3(3 - 3cos^2(x)) + 7cos(x) - 3 = 9 - 9cos^2(x) + 7cos(x) - 3 = -9cos^2(x) + 7cos(x) + 6 = 0
This is now a quadratic equation in terms of cos(x). Let y = cos(x) for convenience:
-9y^2 + 7y + 6 = 0
Now, you can solve this quadratic equation using the quadratic formula or factoring. Once you find the solutions for y (cos(x)), you can then solve for x by taking the arccosine of those values. Remember to verify your solutions within the given range of the trigonometric function.
To solve the equation 3sin^2(x) + 7cos(x) - 3 = 0, we can use some trigonometric identities to simplify the expression.
Start by using the Pythagorean Identity: sin^2(x) + cos^2(x) = 1.
Rewrite the equation in terms of sin^2(x) only
3sin^2(x) = 3 - 3cos^2(x).
Substitute this into the original equation
3(3 - 3cos^2(x)) + 7cos(x) - 3 =
9 - 9cos^2(x) + 7cos(x) - 3 =
-9cos^2(x) + 7cos(x) + 6 = 0
This is now a quadratic equation in terms of cos(x). Let y = cos(x) for convenience:
-9y^2 + 7y + 6 = 0
Now, you can solve this quadratic equation using the quadratic formula or factoring. Once you find the solutions for y (cos(x)), you can then solve for x by taking the arccosine of those values. Remember to verify your solutions within the given range of the trigonometric function.