The trigonometric function ‘Sine' is abbreviated as Sin. Sine is the ratio of the length of a right- angle triangle's perpendicular to its hypotenuse. Sin (a) = height (perpendicular) /hypotenuse or p/h, where a is the angle between the two sides.
The angle formed by the perpendicular and the hypotenuse is known as the sin angle. The angle of 30 degrees between the perpendicular and hypotenuse of a right-angle triangle is Sin 30 degrees. Sin 30 has a predetermined value of 1/2 (in fractional form), which is equal to cos 60 degrees. ‘Secant' or ‘sec' is the inverse of sin.
For sin 30 degrees, there are three basic values. First, as previously stated, the value of a trigonometric ratio is 1/2 in fractional form. Second, the value of sin 30 in the circular system, employing trigonometric functions in the cartesian plane, is pi/6 or 180/6. However, if n number of cycles is present and the result becomes (n x pi/6), this is the specific answer for a single cycle. The last way to determine the sin 30 value is using the centesimal system. Regardless of the approach, sin 30 is always equal to a value of 0.5 in decimal places.
Mathematicians have determined two ways to prove the value of sin 30, namely the proof using properties of the triangle and the practical approach. Let us understand both concepts in detail.
Let us assume a triangle with all the angles equal (equilateral triangle) KLM and let us assume that there are k, l, and m as their respective angles, which measure 60-degrees each. Let the side LM be the base of the triangle. Now, draw a perpendicular on this base from the vertex K. Name the point of intersection as O. Now, there are two triangles KOM and KOL. Taking one triangle, we get angle KOM is 90-degree (angle extended by the perpendicular), and angle KMO is 60-degree (angle of an equilateral triangle). Therefore, using the angle sum property of the triangle: KOM + KMO + MKO = 180, MKO = 180 – 90 + 60, hence, angle MKO equals 30 degrees. Let us assume the sides of the triangle to be 2s. The triangle KOM and KOL are congruent to each other hence LO = OM = 2s/2 = s.
Sine 30 = height of triangle/ hypotenuse
= OL/KL
= s/2s
= ½
We must build a right-angle triangle with one angle of 30 degrees from scratch for the practical proof of the value of sin 30 and then calculate the ratio of length and perpendicularity of the hypotenuse. The following stages are described:
The angle carved from the endpoints of the diameter at any place in the circumference of the semicircle always amounts to 90-degrees. No matter where the point is on the circumference, the angle extended will always be a right angle. This also proves that a right-angled triangle is subtended by a semicircle. The sides other than the base may or may not be equal to each other. They depend on the orientation of the point on the perimeter of the semicircle.