What is sin 60 degree
The Correct Answer and Explanation is:
The sine of 60 degrees is equal to (\frac{\sqrt{3}}{2}) or approximately 0.866. This value is derived from the properties of a 30-60-90 right triangle, which is a special type of triangle in trigonometry. Understanding this value involves exploring the relationship between the angles and the sides of such triangles.
In a 30-60-90 triangle, the sides are in a specific ratio: the side opposite the 30-degree angle is the shortest and is denoted as (x), the side opposite the 60-degree angle is (x\sqrt{3}), and the hypotenuse (the side opposite the right angle) is (2x).
To find (\sin(60^\circ)), we use the definition of the sine function, which is the ratio of the length of the side opposite the angle to the length of the hypotenuse. For the 30-60-90 triangle, if we let the shortest side (opposite the 30-degree angle) be (x), then:
- The side opposite the 60-degree angle (which we need for sine) is (x\sqrt{3}).
- The hypotenuse is (2x).
Thus, we can express (\sin(60^\circ)) as follows:
[
\sin(60^\circ) = \frac{\text{Opposite side}}{\text{Hypotenuse}} = \frac{x\sqrt{3}}{2x}
]
When simplifying this expression, the (x) terms cancel out, resulting in:
[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
]
This value is commonly used in various applications of trigonometry, including in physics, engineering, and geometry. It helps in solving problems related to angles, distances, and even in constructing geometric figures. Moreover, knowing sine values for commonly used angles (0°, 30°, 45°, 60°, and 90°) allows for easier calculations in more complex trigonometric scenarios.