Final answer:
The sin of 30 degrees is proven using an equilateral triangle inscribed in a unit circle, which when divided, forms two 30-60-90 triangles. The side opposite the 30° angle in the unit circle represents the sine value, which geometrically is ½. This proof is a foundational principle in trigonometry and relates to various trigonometric identities.
Step-by-step explanation:
Understanding the Sine of 30 Degrees in a Unit Circle
To understand the sin of 30 degrees, or ½, proof in the context of a unit circle, we can visualize the circle with a radius of 1. A point on the circle that forms a 30° angle with the horizontal axis creates a right triangle within the circle. Using the properties of an equilateral triangle, which the two right triangles together form, we can deduce the length of the sides in terms of sine and cosine.
Consider an equilateral triangle with sides of 2 units inscribed in the unit circle. By drawing a perpendicular from a vertex to the base, we split the equilateral triangle into two 30-60-90 right triangles. The hypotenuse is 2, the side opposite the 30° angle (thus representing the “sin of 30 degrees”) is 1, and the base (opposite the 60° angle) is √3. Since the radius of the unit circle is 1, when scaled down by half, the opposite side to the 30° angle is ½.
Mathematically, the unit circle confirms this since the y-coordinate of a point on the unit circle at an angle θ is equal to sin(θ). At 30°, the y-coordinate is sin(30°), which we found geometrically to be ½. This is a fundamental concept in trigonometry, mirrored in other trigonometric identities and equations such as those mentioned, including the law of sines, the law of cosines, and double angle formulas for sine and cosine.