The quadratic equation x^2−9x+10=0 has roots tanα and tanβ, where 0∘<α+β<180∘. What is the measure of α+β (in degrees)?
x = (9±√41)/2
arctan(x) = 82.6° or 52.4°
Don't know how to evaluate the arctan directly; it appears not to be any familiar angle
To find the measure of α+β, we need to find the values of α and β first. Given that the quadratic equation x^2 - 9x + 10 = 0 has roots tanα and tanβ, we can use the quadratic formula to solve for x.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation x^2 - 9x + 10 = 0, we have:
a = 1
b = -9
c = 10
Plugging these values into the quadratic formula, we get:
x = (-(-9) ± √((-9)^2 - 4(1)(10))) / (2(1))
= (9 ± √(81 - 40)) / 2
= (9 ± √41) / 2
Since the roots of the equation are tangents of angles α and β, we can write:
tanα = (9 + √41) / 2
tanβ = (9 - √41) / 2
To find α and β individually, we can use the inverse tangent function (tan^(-1)) on each root:
α = tan^(-1)((9 + √41) / 2)
β = tan^(-1)((9 - √41) / 2)
Finally, we can calculate the measure of α+β by adding the angles:
α+β = α + β = tan^(-1)((9 + √41) / 2) + tan^(-1)((9 - √41) / 2)
Therefore, the measure of α+β in degrees can be found by evaluating the above expression.