Answer :
To determine the value of [tex]\(\sin 45^\circ\)[/tex], let's go step-by-step through the trigonometric properties and what we know about the sine function.
1. Understanding [tex]\(\sin \theta\)[/tex]:
The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
2. Special Angles:
The angle [tex]\(45^\circ\)[/tex] is one of the special angles in trigonometry, often appearing in isosceles right triangles.
3. 45-45-90 Triangle Properties:
For a right triangle with angles [tex]\(45^\circ-45^\circ-90^\circ\)[/tex], the sides opposite the [tex]\(45^\circ\)[/tex] angles are equal in length. If each leg of the triangle is of length [tex]\(1\)[/tex], the hypotenuse can be found using the Pythagorean theorem:
[tex]\[ \text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2} \][/tex]
4. Calculating [tex]\(\sin 45^\circ\)[/tex]:
Using the definition of sine, the sine of [tex]\(45^\circ\)[/tex] is:
[tex]\[ \sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} \][/tex]
Now, let’s match this result to the given choices:
- A. [tex]\(\frac{1}{2}\)[/tex]
- B. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
- C. [tex]\(\sqrt{2}\)[/tex]
- D. 1
Clearly, the value [tex]\(\sin 45^\circ = \frac{1}{\sqrt{2}}\)[/tex] matches choice B.
Thus, the correct choice is:
B. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
1. Understanding [tex]\(\sin \theta\)[/tex]:
The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
2. Special Angles:
The angle [tex]\(45^\circ\)[/tex] is one of the special angles in trigonometry, often appearing in isosceles right triangles.
3. 45-45-90 Triangle Properties:
For a right triangle with angles [tex]\(45^\circ-45^\circ-90^\circ\)[/tex], the sides opposite the [tex]\(45^\circ\)[/tex] angles are equal in length. If each leg of the triangle is of length [tex]\(1\)[/tex], the hypotenuse can be found using the Pythagorean theorem:
[tex]\[ \text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2} \][/tex]
4. Calculating [tex]\(\sin 45^\circ\)[/tex]:
Using the definition of sine, the sine of [tex]\(45^\circ\)[/tex] is:
[tex]\[ \sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} \][/tex]
Now, let’s match this result to the given choices:
- A. [tex]\(\frac{1}{2}\)[/tex]
- B. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
- C. [tex]\(\sqrt{2}\)[/tex]
- D. 1
Clearly, the value [tex]\(\sin 45^\circ = \frac{1}{\sqrt{2}}\)[/tex] matches choice B.
Thus, the correct choice is:
B. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
