Solve for x please, I simplified to that from 1800 = 2500tanx - 490(1+tan^2x). I think I simplified correctly, but if not solve from the original equation please.
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0.001869 = tanx + tan²x ← Notice that this is a quadratic equation in tanx.
You can solve it by using the Quadratic Formula
or by completing the square.
I'll use completing the square.
0.001869 = tan²x + tanx
0.001869 + (½)² = tan²x + tanx + (½)² ← Added (b/2)² to complete the square
0.001869 + 0.25 = (tanx + ½)²
(tanx + 0.5)² = 0.251869 ← Now, take the square root of both sides
tanx + 0.5 = ±√[0.251869] ← By the square root property
tanx = -0.5 ± √[0.251869]
tanx = 0.001866 or -1.00187
So, for 0 ≤ x ≤ 2π:
x = arctan(0.001866) = 0.001866 ✔ or (π + 0.001866) = 3.14346 ✔
OR
x = arctan(-1.00187) = -0.786332 or (π - 0.786332)
2π-0.786332 or (π - 0.786332)
5.49685 ✔ or 2.35526 ✔
ANSWER
x ≈ 0.001866, 2.35526, 3.14346, 5.49685
Hope that helps
——————————————————————————————————————
0.001869 = tanx + tan²x ← Notice that this is a quadratic equation in tanx.
You can solve it by using the Quadratic Formula
or by completing the square.
I'll use completing the square.
0.001869 = tan²x + tanx
0.001869 + (½)² = tan²x + tanx + (½)² ← Added (b/2)² to complete the square
0.001869 + 0.25 = (tanx + ½)²
(tanx + 0.5)² = 0.251869 ← Now, take the square root of both sides
tanx + 0.5 = ±√[0.251869] ← By the square root property
tanx = -0.5 ± √[0.251869]
tanx = 0.001866 or -1.00187
So, for 0 ≤ x ≤ 2π:
x = arctan(0.001866) = 0.001866 ✔ or (π + 0.001866) = 3.14346 ✔
OR
x = arctan(-1.00187) = -0.786332 or (π - 0.786332)
2π-0.786332 or (π - 0.786332)
5.49685 ✔ or 2.35526 ✔
ANSWER
x ≈ 0.001866, 2.35526, 3.14346, 5.49685
Hope that helps
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