Let z be a complex number such that z = 2(cos 8∘ + i cos 82∘).Then z^5 can be expressed as r(sin α∘+ i cos α∘), where r is a real number and 0 ≤ α ≤ 90. What is the value of r+α?

Hint to solve:

Example Question:

Let z be a complex number such that z = 2(cos 5∘ + i cos 85∘). Then z^6 can be expressed as r(sin α∘ + i cos α∘), where r is a real number and 0 ≤ α ≤90. What is the value of r+α?

Solution:

Since cos85∘=cos(90∘−5∘)=sin5∘,z=2(cos5∘+icos85∘)=2(cos5∘+isin5∘).

Applying De Moivre's formula gives

z^6=26(cos(6⋅5∘)+isin(6⋅5∘))=64(cos30∘+isin30∘).
Since cos30∘=cos(90∘−60∘)=sin60∘ and sin30∘=sin(90∘−60∘)=cos60∘,

z^6=64(cos30∘+isin30∘)=64(sin60∘+icos60∘).
Therefore, r=64 and α=60, hence r + α = 64 + 60 = 124.

To solve the given problem, we need to apply De Moivre's formula.

De Moivre's formula states that for any complex number z = r(cosθ + i sinθ), the nth power of z can be expressed as:

z^n = r^n(cos(nθ) + i sin(nθ))

In this case, we are given that z = 2(cos8∘ + i cos82∘).

To find z^5, we can directly apply De Moivre's formula:

z^5 = 2^5(cos(5(8∘)) + i sin(5(8∘)))

Simplifying inside the parentheses:

z^5 = 32(cos(40∘) + i sin(40∘))

Now we have the expression for z^5 in the form r(sinα∘ + i cosα∘). We need to determine the values of r and α.

From the given expression, r = 32.

To find α, we need to determine the angle whose cosine is cos(40∘).

Since 0 ≤ α ≤ 90, we know that α is an acute angle. Therefore, we can use the identity cos(α) = sin(90 - α) to find α.

cos(40∘) = sin(90 - 40∘) = sin(50∘)

So, α = 50∘.

The value of r + α is then:

r + α = 32 + 50 = 82.

Therefore, the value of r + α is 82.