Sure, let's go through each part step-by-step:
(i) [tex]\(58^\circ\)[/tex]
The complement of an angle [tex]\(A\)[/tex] is given by [tex]\(90^\circ - A\)[/tex].
So, for an angle of [tex]\(58^\circ\)[/tex]:
[tex]\[
90^\circ - 58^\circ = 32^\circ
\][/tex]
Thus, the complement of [tex]\(58^\circ\)[/tex] is [tex]\(32^\circ\)[/tex].
(ii) [tex]\(160^\circ\)[/tex]
An angle greater than [tex]\(90^\circ\)[/tex] does not have a complement, as the definition of complementary angles requires that the two angles add up to [tex]\(90^\circ\)[/tex].
Therefore, there is no complement for [tex]\(160^\circ\)[/tex].
(iii) [tex]\(\frac{2}{3}\)[/tex] of a right angle
First, we need to find what [tex]\(\frac{2}{3}\)[/tex] of a right angle is. Since a right angle is [tex]\(90^\circ\)[/tex]:
[tex]\[
\frac{2}{3} \times 90^\circ = 60^\circ
\][/tex]
Now, we need to find the complement of [tex]\(60^\circ\)[/tex].
[tex]\[
90^\circ - 60^\circ = 30^\circ
\][/tex]
Thus, the complement of [tex]\(\frac{2}{3}\)[/tex] of a right angle is [tex]\(30^\circ\)[/tex].
Summary of Results:
1. The complement of [tex]\(58^\circ\)[/tex] is [tex]\(32^\circ\)[/tex].
2. There is no complement for [tex]\(160^\circ\)[/tex] as it is greater than [tex]\(90^\circ\)[/tex].
3. The complement of [tex]\(\frac{2}{3}\)[/tex] of a right angle is [tex]\(30^\circ\)[/tex].
