Sin‌ ‌45‌ ‌value‌

Sine is the ratio of the length of the perpendicular side of a right-angled triangle to its hypotenuse. Sine is considered an important trigonometric function, other than cosine and tangent functions.

Consider a triangle ABC with a and b as its legs and h as the hypotenuse. Consider both the other angles as 45 degrees. In order to find the value of sin 45, let us assume the length of both the legs of a unit value.

Therefore, a = b = 1

From the Pythagoras theorem, h² = a² + b²

We get, h = √2

From the definition of a sine angle, sin 45° = perpendicular/hypotenuse

= a/h = 1/√2 = 0.707

Other ratios of sine

Values of other trigonometric ratios

Importance of sine function

1. The sine function is essential because it is the only periodic function whose wave shape retains its shape when added to another wave of the same frequency, phase, and magnitude.
2. Sine makes the Fourier series acoustically unique.
3. The sine function is used in periodic phenomena, like average temperature variations throughout the year, sound, waves, the velocity of harmonic oscillators, and sunlight intensity and day length.

Properties relating to the quadrants

Laws of sines

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:

sin A/a = sin B/b = sin C/c

This is equivalent to the equality of the first three expressions below:

a/sin A = b/sin B = c/sin C = 2R

Where, R is the triangle's circumradius.

Fun facts

1. The word sine comes from the Latin mistranslation of Arabic jiba by Robert of Chester, a transliteration of the Sanskrit word for the half chord, jyaardha.
2. Sine function was traced during the Gupta period from the famous books Aryabhatiya and Surya Sidhanta.
3. Sine wave is the most used trigonometric function for generating waves and performing scientific calculations.