Can anyone help me with these tasks?

Do you have an equation like

sin(x) = y

given, then all real solutions, G = ℝ, are given by the expressions

⌈ x = –sin⁻¹(y) + π (2 n + 1)

⌊ x = sin⁻¹(y) + 2 π n

with any integers n.

Do you have an equation like

cos(x) = y

given, then all real solutions, G = ℝ, are given by the expressions

⌈ x = –cos⁻¹(y) + 2 π n

⌊ x = cos⁻¹(y) + 2 π n

with any integers n.

Task 1)

a)

sin(x) = –0.4

Using the formulas above we get:

⌈ x = –sin⁻¹(y) + π (2 n + 1)

⌊ x = sin⁻¹(y) + 2 π n

⌈ x = –(–0.41) + π (2 n + 1)

⌊ x = –0.41 + 2 π n

⌈ x ≈ 0.41 + π (2 n + 1)

⌊ x ≈ –0.41 + 2 π n

With G = [–4, 4] we get:

For the upper x we ​​get:

⌈ x ≈ –9.01 (n=–2) ∉ G

∣ x ≈ –2.73 (n=–1) ∈ G

∣ x ≈ 3.55 (n=0) ∈ G

⌊ x ≈ 9.83 (n=1) ∉ G

For the value below x we ​​get:

⌈ x ≈ –6.69 (n=–1) ∉ G

∣ x ≈ –0.41 (n=0) ∈ G

⌊ x ≈ 5.87 (n=1) ∉ G

Other numbers n need not be considered, since x then lie even further away from G. The solution set is therefore

L = {–2.73; –0.41; 3.55}.

b)

cos(x) = –0.8

Using the formulas above we get:

⌈ x = –cos⁻¹(y) + 2 π n

⌊ x = cos⁻¹(y) + 2 π n

⌈ x ≈ –1.98 + 2 π n

⌊ x ≈ 1.98 + 2 π n

With G = [–4, 4] we get:

For the upper x we ​​get:

⌈ x ≈ –8.26 (n=–1) ∉ G

∣ x ≈ –1.98 (n=0) ∈ G

⌊ x ≈ 4.30 (n=1) ∉ G

For the value below x we ​​get:

⌈ x ≈ –4.30 (n=–1) ∉ G

∣ x ≈ 1.98 (n=0) ∈ G

⌊ x ≈ 8.26 (n=1) ∉ G

Other numbers n need not be considered, since x then lie even further away from G. The solution set is therefore

L = {–1.98; 1.98}.

Task 2)

Look at the formulas for the cosine equation above. Using them (we're looking for zeros of the cosine function, so y = 0):

cos(x) = 0

⌈ x = –cos⁻¹(0) + 2 π n

⌊ x = cos⁻¹(0) + 2 π n

⌈ x = –π/2 + 2 π n

⌊ x = π/2 + 2 π n

Since G = ℝ, there is no restriction. The solution set is

L = {–π/2 + 2π n | n ∈ ℤ} ∪ {π/2 + 2 π n | n ∈ ℤ}.

If you don't know: "∪" means "unite".

Task 3)

Plot the cosine function in a coordinate system. Next, draw the horizontal line y = 0.6. Now mark the intersection points of the two graphs. Read the x-coordinate from these points and check whether x is in the interval [3, 12) or whether 3 ≤ x < 12. All x in this interval are solutions, and they are summarized as a solution set. The sketch looks like this (the interval is in green):

There are therefore exactly three intersections in the interval. If you read them off, you get approximately the solution set

L = {5.35; 7.20; 11.60}.