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Solution inequality：
Directly input the univariate inequality (that is, the inequality containing only one variable), set the angular unit (radian or angle) of the trigonometric function, and click the "Next" button to obtain the solution set of the inequality.
It supports mathematical functions (including trigonometric functions).
Directly input the univariate inequality (that is, the inequality containing only one variable), set the angular unit (radian or angle) of the trigonometric function, and click the "Next" button to obtain the solution set of the inequality.
It supports mathematical functions (including trigonometric functions).
Current location：Mathematical operation > History of Inequality Computation > Answer
Overview: 1 questions will be solved this time.Among them
☆1 inequalities
[ 1/1Inequality]
Assignment:Find the solution set of inequality √{{k+√[k^2+(sin45)^2-2*k*(sin45)^2]}^2+k^2-2*k*{k+√[k^2+(sin45)^2-2*k*(sin45)^2]*cos90}}
Solution:
The inequality can be reduced to 1 inequality：
√ ( ( k + √ ( k ^ 2 + ( sin 45 ) ^ 2 - 2 * k * ( sin 45 ) ^ 2 ) ) ^ 2 + k ^ 2 - 2 * k * ( k + √ ( k ^ 2 + ( sin 45 ) ^ 2 - 2 * k * ( sin 45 ) ^ 2 ) * cos 90 ) ) < cos 45 + 1 - k (1)
From the definition field of √
x ^ 2 + ( sin 45 ) ^ 2 - 2 * x * ( sin 45 ) ^ 2 ≥ 0 (2 )
From the definition field of √
x ^ 2 + ( sin 45 ) ^ 2 - 2 * x * ( sin 45 ) ^ 2 ≥ 0 (3 )
From the definition field of √
( x + √ ( x ^ 2 + ( sin 45 ) ^ 2 - 2 * x * ( sin 45 ) ^ 2 ) ) ^ 2 + x ^ 2 - 2 * x * ( x + √ ( x ^ 2 + ( sin 45 ) ^ 2 - 2 * x * ( sin 45 ) ^ 2 ) * cos 90 ) ≥ 0 (4 )
From inequality(1)：
k < 0.532176
From inequality(2)：
k ∈ R (R is all real numbers),that is, in the real number range, the inequality is always established!
From inequality(3)：
k ∈ R (R is all real numbers),that is, in the real number range, the inequality is always established!
From inequality(4)：
k ≥ -0.426048
From inequalities (1) and (2)
k < 0.532176 (5)
From inequalities (3) and (5)
k < 0.532176 (6)
From inequalities (4) and (6)
-0.426048 ≤ k < 0.532176 (7)
The final solution set is :
-0.426048 ≤ k < 0.532176