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.
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.
