1. sin^2 theta+cos theta=2 (Hint: Use the Pythagorean identity sin^2 theta+cos theta=1 to replace sin^2 theta in the given equation.)
I got no solution
sin^2 Ø + cosØ -2 = 0
1-cos^2 Ø + cosØ - 2 = 0
cos^2 Ø - cosØ + 1 = 0
cosØ = (1 ± √-3)/2
which is not real
you are correct, there is no solution.
Let's solve the equation step by step:
1. Given equation: sin^2(theta) + cos(theta) = 2
2. Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can replace sin^2(theta) with 1 - cos^2(theta):
(1 - cos^2(theta)) + cos(theta) = 2
3. Distributing the negative sign:
1 - cos^2(theta) + cos(theta) = 2
4. Rearranging the equation:
-cos^2(theta) + cos(theta) + 1 = 2
5. Moving all terms to one side of the equation:
-cos^2(theta) + cos(theta) - 1 = 0
6. Let's rewrite the equation as a quadratic equation:
cos^2(theta) - cos(theta) + 1 = 0
7. Applying the quadratic formula to solve for cos(theta):
cos(theta) = [-(-1) ± sqrt((-1)^2 - 4(1)(1))] / (2(1))
cos(theta) = [1 ± sqrt(1 - 4)] / 2
cos(theta) = [1 ± sqrt(-3)] / 2
8. Since the square root of a negative number is not a real number, there are no real solutions to this equation. Thus, there is no solution for theta that satisfies the given equation sin^2(theta) + cos(theta) = 2.
To solve the given equation, we can use the Pythagorean identity as the hint suggests. The Pythagorean identity states that sin^2(theta) + cos^2(theta) = 1.
We are given the equation sin^2(theta) + cos(theta) = 2, and we can replace sin^2(theta) with 1 - cos^2(theta) using the Pythagorean identity.
The equation becomes:
1 - cos^2(theta) + cos(theta) = 2
Rearranging the terms:
cos^2(theta) - cos(theta) + 1 = 0
Now, we can solve this quadratic equation for cos(theta) using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. In this case, a = 1, b = -1, and c = 1.
Applying the quadratic formula:
cos(theta) = [1 ± √((-1)^2 - 4(1)(1))] / 2(1)
cos(theta) = [1 ± √(1 - 4)] / 2
cos(theta) = [1 ± √(-3)] / 2
Since the square root of a negative number is not a real number, there are no real solutions to this equation.
Therefore, the equation sin^2(theta) + cos(theta) = 2 has no solution.