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cos^2(x)+cos^2(30)+cos^2(75)=1

Решение

cos^{2} (x) + cos^{2} (30) + cos^{2} (75) = 1

x = 1.20704 \dots + 2 πn, x = 2 π - 1.20704 \dots + 2 πn, x = 1.93454 \dots + 2 πn, x = - 1.93454 \dots + 2 πn

Градусы

x = 69.15859 \dots^{\circ} + 36 0^{\circ} n, x = 290.84140 \dots^{\circ} + 36 0^{\circ} n, x = 110.84140 \dots^{\circ} + 36 0^{\circ} n, x = - 110.84140 \dots^{\circ} + 36 0^{\circ} n

Шаги решения

cos^{2} (x) + cos^{2} (30) + cos^{2} (75) = 1

Решитe подстановкой

cos^{2} (x) + cos^{2} (30) + cos^{2} (75) = 1

Допустим:

cos (x) = u

u^{2} + cos^{2} (30) + cos^{2} (75) = 1

u^{2} + cos^{2} (30) + cos^{2} (75) = 1 : u = 1 - cos^{2} (30) - cos^{2} (75), u = - 1 - cos^{2} (30) - cos^{2} (75)

u^{2} + cos^{2} (30) + cos^{2} (75) = 1

Переместите

cos^{2} (30)

вправо

u^{2} + cos^{2} (30) + cos^{2} (75) = 1

Вычтите

cos^{2} (30)

с обеих сторон

u^{2} + cos^{2} (30) + cos^{2} (75) - cos^{2} (30) = 1 - cos^{2} (30)

После упрощения получаем

u^{2} + cos^{2} (75) = 1 - cos^{2} (30)

u^{2} + cos^{2} (75) = 1 - cos^{2} (30)

Переместите

cos^{2} (75)

вправо

u^{2} + cos^{2} (75) = 1 - cos^{2} (30)

Вычтите

cos^{2} (75)

с обеих сторон

u^{2} + cos^{2} (75) - cos^{2} (75) = 1 - cos^{2} (30) - cos^{2} (75)

После упрощения получаем

u^{2} = 1 - cos^{2} (30) - cos^{2} (75)

u^{2} = 1 - cos^{2} (30) - cos^{2} (75)

Для

x^{2} = f (a)

решениями являются

x = f (a), - f (a)

u = 1 - cos^{2} (30) - cos^{2} (75), u = - 1 - cos^{2} (30) - cos^{2} (75)

Делаем обратную замену

u = cos (x)

cos (x) = 1 - cos^{2} (30) - cos^{2} (75), cos (x) = - 1 - cos^{2} (30) - cos^{2} (75)

cos (x) = 1 - cos^{2} (30) - cos^{2} (75), cos (x) = - 1 - cos^{2} (30) - cos^{2} (75)

cos (x) = 1 - cos^{2} (30) - cos^{2} (75) : x = arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = 2 π - arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

cos (x) = 1 - cos^{2} (30) - cos^{2} (75)

Примените обратные тригонометрические свойства

cos (x) = 1 - cos^{2} (30) - cos^{2} (75)

Общие решения для

cos (x) = 1 - cos^{2} (30) - cos^{2} (75)

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = 2 π - arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

x = arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = 2 π - arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

cos (x) = - 1 - cos^{2} (30) - cos^{2} (75) : x = arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = - arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

cos (x) = - 1 - cos^{2} (30) - cos^{2} (75)

Примените обратные тригонометрические свойства

cos (x) = - 1 - cos^{2} (30) - cos^{2} (75)

Общие решения для

cos (x) = - 1 - cos^{2} (30) - cos^{2} (75)

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = - arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

x = arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = - arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

Объедините все решения

x = arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = 2 π - arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = - arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

Покажите решения в десятичной форме

x = 1.20704 \dots + 2 πn, x = 2 π - 1.20704 \dots + 2 πn, x = 1.93454 \dots + 2 πn, x = - 1.93454 \dots + 2 πn