Sin 30 value

General
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sin 30 value

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What is the value of sin 30?

Answer:
The value of \sin(30^\circ) is a fundamental trigonometric identity that is often introduced in high school mathematics. Here, 30 degrees is a special angle in trigonometry, and its sine value is straightforward to derive and remember.

Solution By Steps:

  1. Special Right Triangle (30-60-90 Triangle):

    • To understand why \sin(30^\circ) = 0.5, consider a 30-60-90 triangle. This is a special right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees.
    • In a 30-60-90 triangle, the sides are in the ratio 1:√3:2. This means that if the hypotenuse (side opposite the 90-degree angle) is 2, the side opposite the 30-degree angle (shortest side) is 1, and the side opposite the 60-degree angle is \sqrt{3}.

  2. Sine Definition:

    • The sine of an angle in a right triangle is defined as the length of the side opposite the angle divided by the length of the hypotenuse.

      \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

  3. Applying the Definition:

    • For a 30-degree angle in our 30-60-90 triangle:

      • The length of the side opposite to the 30-degree angle (opposite) is 1.
      • The length of the hypotenuse is 2.

    • Therefore, the sine of 30 degrees is:

      \sin(30^\circ) = \frac{1}{2} = 0.5

Confirmation with Unit Circle:

  • On the unit circle, the coordinates of a point corresponding to a 30-degree angle from the positive x-axis are (\sqrt{3}/2, 1/2). The y-coordinate of this point represents the sine value.
  • Thus, \sin(30^\circ) = 1/2 = 0.5.

Final Answer:
The value of \sin(30^\circ) is \boxed{0.5}.