Answer:
Here is the answer
Explanation:
The quadratic equation 5x² - 3x + c = 0 has roots that are the values of x that make the equation equal to 0. In this case, we are asked to find the values of c such that the roots of the equation are sin(a) and cos(a).
To find the roots of a quadratic equation, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 5, b = -3, and c is the unknown value that we are trying to find.
Substituting these values into the quadratic formula, we get:
x = (-(-3) ± √((-3)² - 4 * 5 * c)) / (2 * 5)
= (3 ± √(9 - 20c)) / 10
If the roots of the equation are sin(a) and cos(a), then we must have:
x = sin(a) and x = cos(a)
Substituting these expressions into the formula for x, we get:
sin(a) = (3 ± √(9 - 20c)) / 10
cos(a) = (3 ± √(9 - 20c)) / 10
We can solve this system of equations by setting the two expressions for x equal to each other and solving for c:
sin(a) = cos(a)
(3 ± √(9 - 20c)) / 10 = (3 ± √(9 - 20c)) / 10
This equation is true for any value of c, so the roots of the quadratic equation 5x² - 3x + c = 0 are sin(a) and cos(a) for any value of c.