Question

Cosп/4×sinx-sinп/4×cosx<-3/2​

Answer 1

Ответ:Формула:

sin α ·cos β –cos α ·sin β =sin( α – β )

sin(x–(π/4))=√3/2

Уравнение: sint=√3/2 – простейшее тригонометрическое уравнение решают по формулам: t=(–1)karcsin(√3/2)+πk, k ∈ Z

х–(π/4)=(–1)karcsin(√3/2)+πk, k ∈ Z

х–(π/4)=(–1)k·(π/3)+πk, k ∈ Z

х=(–1)k·(π/3)+(π/4)+πk, k ∈ Z – это ответ.

Так как (–1)k·(π/3)+πk, k ∈ Z можно записать в виде серии из двух ответов:

k=2n

(π/3)+2πn, n ∈ Z

k=2n+1

(2π/3)+2πn, n ∈ Z

то ответ можно записать и так.

х=(π/3)+(π/4)+2πn=(7π/12)+2πn, n ∈ Z или

х=(2π/3)+(π/4)+2πn=(11π/12)+2πn, n ∈ Z

Такая запись полезна при отборе корней

Объяснение:

Answer 2

To solve the inequality cos(π/4) * sin(x) - sin(π/4) * cos(x) < -3/2, you can use trigonometric identities to simplify it and find its solution.

First, let's use the trigonometric identity sin(a - b) = sin(a)cos(b) - cos(a)sin(b) to rewrite the left side of the inequality:

sin(π/4 - x) < -3/2

Now, let's work on solving this inequality. First, find the values of x that satisfy sin(π/4 - x) = -3/2. However, sin values are limited to the range [-1, 1], so there are no real solutions for sin(π/4 - x) = -3/2.

Therefore, the inequality cos(π/4) * sin(x) - sin(π/4) * cos(x) < -3/2 has no real solutions.

In other words, there are no real values of x that make the inequality true.