Answer :
To find the value of \(\sin 45^{\circ}\), let's use the properties of trigonometry:
1. Understanding the Angle: The angle \(45^{\circ}\) is special in trigonometry. It is one of the standard angles, coming from an \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle.
2. Special Triangle Properties:
- In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the sides opposite the \(45^{\circ}\) angles are equal.
- If we assume each of these sides has a length of \(1\), then the length of the hypotenuse (by the Pythagorean theorem) will be:
[tex]\[ \text{hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2} \][/tex]
3. Sine Definition:
- The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
- Thus, for \(\sin 45^{\circ}\):
[tex]\[ \sin 45^{\circ} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} \][/tex]
4. Rationalizing the Denominator:
- To express this without a square root in the denominator, we multiply the numerator and the denominator by \(\sqrt{2}\):
[tex]\[ \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
So, the value of \(\sin 45^{\circ}\) is \(\frac{\sqrt{2}}{2}\).
When you calculate the decimal value of \(\frac{\sqrt{2}}{2}\), you get approximately \(0.7071067811865475\).
Hence, the correct answer matches the numerical value derived, and among the given options:
A. \(\sqrt{2}\)
B. \(\frac{1}{2}\)
C. \(\frac{1}{\sqrt{2}}\)
D. 1
The answer is:
C. [tex]\(\frac{1}{\sqrt{2}}\)[/tex].
1. Understanding the Angle: The angle \(45^{\circ}\) is special in trigonometry. It is one of the standard angles, coming from an \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle.
2. Special Triangle Properties:
- In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the sides opposite the \(45^{\circ}\) angles are equal.
- If we assume each of these sides has a length of \(1\), then the length of the hypotenuse (by the Pythagorean theorem) will be:
[tex]\[ \text{hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2} \][/tex]
3. Sine Definition:
- The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
- Thus, for \(\sin 45^{\circ}\):
[tex]\[ \sin 45^{\circ} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} \][/tex]
4. Rationalizing the Denominator:
- To express this without a square root in the denominator, we multiply the numerator and the denominator by \(\sqrt{2}\):
[tex]\[ \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
So, the value of \(\sin 45^{\circ}\) is \(\frac{\sqrt{2}}{2}\).
When you calculate the decimal value of \(\frac{\sqrt{2}}{2}\), you get approximately \(0.7071067811865475\).
Hence, the correct answer matches the numerical value derived, and among the given options:
A. \(\sqrt{2}\)
B. \(\frac{1}{2}\)
C. \(\frac{1}{\sqrt{2}}\)
D. 1
The answer is:
C. [tex]\(\frac{1}{\sqrt{2}}\)[/tex].
