What is sin 60°?
1
60
90°
√3
2
30
The Correct Answer and Explanation is :
The value of (\sin 60^\circ) is (\frac{\sqrt{3}}{2}).
Explanation:
The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. The angles commonly used in trigonometry include 0°, 30°, 45°, 60°, and 90°. To understand the sine of 60°, we can refer to a 30-60-90 triangle, which is a special type of right triangle.
In a 30-60-90 triangle, the ratios of the sides are well-defined:
- The side opposite the 30° angle (the shortest side) is half the length of the hypotenuse.
- The side opposite the 60° angle (the longer leg) is (\sqrt{3}) times the length of the side opposite the 30° angle.
If we let the hypotenuse be 2, then the lengths of the sides can be determined as follows:
- The length of the side opposite the 30° angle is (1) (which is (\frac{1}{2} \times 2)).
- The length of the side opposite the 60° angle is (\sqrt{3}) (which is (\sqrt{3} \times 1)).
Now, we can calculate (\sin 60^\circ):
[
\sin 60^\circ = \frac{\text{Opposite Side}}{\text{Hypotenuse}} = \frac{\text{Length of side opposite } 60^\circ}{\text{Length of Hypotenuse}} = \frac{\sqrt{3}}{2}.
]
This relationship holds true in the unit circle as well, where the coordinates of a point corresponding to an angle (\theta) give the values of (\cos \theta) and (\sin \theta). At (60^\circ), the coordinates are (\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)), confirming that (\sin 60^\circ = \frac{\sqrt{3}}{2}).
In summary, (\sin 60^\circ = \frac{\sqrt{3}}{2}) is derived from the properties of a 30-60-90 triangle and is a critical value in trigonometry used in various applications, including physics and engineering.