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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

Sine 0° 0
Sine 30° or Sine π/6 1/2
Sine 45° or Sine π/4 1/√2
Sine 60°or Sine π/3 √3/2
Sine 90° or Sine π/2 1
Sine 120° or Sine 2π/3 √3/2
Sine 150° or Sine 5π/6 1/2
Sine 180° or Sine π 0
Sine 270° or Sine 3π/2 -1
Sine 360°or Sine 2π 0

Values of other trigonometric ratios

Degree 0 30 45 60 90 180 270 360
Sin 0 1/2 1/√2 √3/2 1 0 -1 0
Cos 1 √3/2 1/√2 1/2 0 -1 0 1
Tan 0 1/√3 1 √3 0 0

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

Quadrant Degrees Radians Value Sign Monotony Convexity
1st quadrant, I 0° < x < 90° 0 < x < π/2 0 < sin x < 1 + increasing concave
2nd quadrant, II 90° < x < 180° π/2 < x < π 0 < sin x < 1 + decreasing concave
3rd quadrant, III 180° < x < 270° π < x < 3π/2 -1 < sin x < 0 - decreasing convex
4th quadrant, IV 270° < x < 360° 3π/2 < x < 2π -1 < sin x < 0 - increasing convex

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.
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