If [tex]$\sin 30^{\circ}$[/tex] is [tex]$\frac{1}{2}$[/tex], then the [tex]$\cos 30^{\circ}$[/tex] equals:

A. [tex]$60 \div \frac{1}{2}$[/tex]
B. [tex]$30: \frac{\sqrt{2}}{2}$[/tex]
C. [tex]$60 \div \frac{\sqrt{3}}{2}$[/tex]
D. [tex]$30^{\circ}: 1$[/tex]



Answer :

Certainly! Let’s analyze and solve this problem step-by-step.

1. Cosine of 60 Degrees:

We know that:
[tex]\[\sin 30^\circ = \frac{1}{2}\][/tex]

Using the complementary angle identity for sine and cosine, which states:
[tex]\[\cos 60^\circ = \sin 30^\circ\][/tex]

We have:
[tex]\[\cos 60^\circ = \frac{1}{2}\][/tex]

2. Evaluating the Given Expressions:

Let's evaluate and compare each of the given mathematical expressions:

- Expression 1: [tex]\(60 \div \frac{1}{2}\)[/tex]

Division by a fraction is equivalent to multiplication by its reciprocal. Thus:
[tex]\[ 60 \div \frac{1}{2} = 60 \times 2 = 120 \][/tex]

- Expression 2: [tex]\(30 \div \frac{\sqrt{2}}{2}\)[/tex]

Here we're dividing 30 by the fraction [tex]\(\frac{\sqrt{2}}{2}\)[/tex], so:
[tex]\[ 30 \div \frac{\sqrt{2}}{2} = 30 \times \frac{2}{\sqrt{2}} = 30 \times \frac{2}{\sqrt{2}} = 30 \times \frac{2\sqrt{2}}{2} = 30 \sqrt{2} = 30 \times 1.4142 \approx 42.4264 \][/tex]

- Expression 3: [tex]\(60 \div \frac{\sqrt{3}}{2}\)[/tex]

Similarly, dividing 60 by [tex]\(\frac{\sqrt{3}}{2}\)[/tex] gives:
[tex]\[ 60 \div \frac{\sqrt{3}}{2} = 60 \times \frac{2}{\sqrt{3}} = 60 \times \frac{2\sqrt{3}}{3} = \frac{120}{\sqrt{3}} = \frac{120\sqrt{3}}{3} = 40 \sqrt{3} = 40 \times 1.7320 \approx 69.2820 \][/tex]

- Expression 4: [tex]\(30^\circ : 1\)[/tex]

This means just comparing the number 30 with 1:
[tex]\[ 30 = 1 \][/tex]

3. Summary of Results:

Let's compile the evaluated results:
- [tex]\(\cos 60^\circ = 0.5\)[/tex]
- [tex]\(60 \div \frac{1}{2} = 120\)[/tex]
- [tex]\(30 \div \frac{\sqrt{2}}{2} \approx 42.4264\)[/tex]
- [tex]\(60 \div \frac{\sqrt{3}}{2} \approx 69.2820\)[/tex]
- [tex]\(30^\circ : 1 = 1\)[/tex]

So the matching results are:
- [tex]\(\cos 60^\circ = 0.5\)[/tex]

Therefore, amongst the expressions given for comparison, there is no match for exactly [tex]\( \cos 60^\circ \)[/tex], however, we calculated logical results for each of them.