In right triangle trigonometry, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse. For a 30-degree angle in a right triangle, this ratio is always 1/2, regardless of the size of the triangle.
Here’s why:
Consider a right triangle with a 30-degree angle. The sides of this triangle are in a specific ratio. The side opposite the 30-degree angle is half the length of the hypotenuse. This is a property of 30-60-90 triangles, where the sides are in the ratio 1 : √3 : 2.
Let’s break it down:
- The side opposite the 30-degree angle is 1 unit.
- The hypotenuse is 2 units.
- The side opposite the 60-degree angle is √3 units.
Since sin(θ) = opposite / hypotenuse, for θ = 30°:
sin(30°) = opposite / hypotenuse = 1 / 2
This ratio remains constant because the sides of the triangle scale proportionally. If you double the size of the triangle, both the opposite side and the hypotenuse will double, keeping the ratio 1/2.
Therefore, sin 30° is always 1/2, no matter how large or small the triangle is.