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At what values ​​of c are the numbers sin(a) and cos(a) the roots of the quadratic equation 5x² - 3x + c = 0 (a is some angle)?

User Cen
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1 Answer

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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.

User Urbz
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