How to Find sin 30 Without a Calculator?

You can find sin 30 without a calculator by understanding the properties of a 30-60-90 right triangle.

Using the 30-60-90 Triangle

Here's how to determine sin 30:

1. Visualize a 30-60-90 triangle: This is a special right triangle with angles of 30°, 60°, and 90°.

2. Side Ratios: The sides of a 30-60-90 triangle have a specific ratio. If the side opposite the 30° angle (the shortest side) has a length of 1, then:

   • The side opposite the 30° angle (shortest side) = 1
   • The side opposite the 60° angle = √3
   • The hypotenuse (opposite the 90° angle) = 2

3. Sine Definition: Remember that sine (sin) 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.

   • sin(angle) = Opposite / Hypotenuse

4. Calculate sin 30: For a 30° angle in our 30-60-90 triangle:

   • Opposite side = 1
   • Hypotenuse = 2

   Therefore, sin 30° = 1 / 2 = 0.5

In summary, sin 30 = 1/2 because in a 30-60-90 triangle, the ratio of the side opposite the 30-degree angle to the hypotenuse is 1:2.