Sine or the **sine function** is one of the three primary functions in trigonometry, the others being cosine, and tan functions. The sine x or sine theta can be defined as the ratio of the opposite side of a right triangle to its hypotenuse. Here, a detailed lesson on this trigonometric function i.e. the sine function is given which will include the following topics:

- Definition
- Formula
- Table of Sine Values
- Properties
- Graph
- Inverse Sine
- Sine Identities
- Sine Calculus
- Law of Sine
- Worksheet

## Sine Function Definition

The sine function can be defined as the ratio of the length of the opposite side to that of the hypotenuse in a right angled triangle. For any triangle, suppose ABC, with an angle alpha, the sine function will be:

## Sine Function Formula

From the above diagram, it is known that Sin α = BC/AB.

Now, the sine formula can be written as:

Sin α = a/h

**Other Trigonometric Functions:**

## Sine Values Table

Sine Degrees |
Values |

Sine 0° | 0 |

Sine 30° | 1/2 |

Sine 45° | 1/√2 |

Sine 60° | √3/2 |

Sine 90° | 1 |

Sine 120° | √3/2 |

Sine 150° | 1/2 |

Sine 180° | 0 |

Sine 270° | -1 |

Sine 360° | 0 |

### Additional Sine Values

Sine 1 Degree is 0.84 | Sine 2 Degree is 0.91 |

Sine 5 Degree is -0.96 | Sine 10 Degree is -0.54 |

Sine 20 Degree is 0.91 | Sine 30 Degree is -0.99 |

Sine 40 Degree is 0.75 | Sine 50 Degree is -0.26 |

Sine 70 Degree is 0.77 | Sine 80 Degree is -0.99 |

Sine 100 Degree is -0.50 | Sine 105 Degree is -0.97 |

Sine 210 Degree is 0.47 | Sine 240 Degree is 0.95 |

Sine 330 Degree is -0.13 | Sine 350 Degree is 0.95 |

## Properties of Sine as per Quadrants

The sine function has values positive or negative depending upon the quadrants. In the above table, it is seen that sine 270 is negative while sine 90 is positive. For the sine function, the value depends upon the quadrants and is positive for first and second quadrants while it is negative for values in third and fourth quadrants.

**Four quadrants in Trigonometry diagram:**

Degree Range |
Quadrant |
Sine Function Sign |
Sine Value Range |

0 to 90 Degrees | 1st Quadrant | + (Positive) | 0 < sin(x) < 1 |

90 to 180 Degrees | 2nd Quadrant | + (Positive) | 0 < sin(x) < 1 |

180 to 270 Degrees | 3rd Quadrant | – (Negative) | -1 < sin(x) < 0 |

270 to 360 Degrees | 4th Quadrant | – (Negative) | -1 < sin(x) < 0 |

## Sine Function Graph

The sine graph looks like the image given below. The **sine graph** or sinusoidal graph is an up-down graph and repeats every 360 degrees i.e. at 2π. In the below-given diagram, it can be seen that from 0, the sine graph rises till +1 and then falls back till -1 from where it rises again.

### Inverse Sine (arc sine)

The sine inverse function is used to measure the angle of an right angled triangle from the given ratios. The inverse of sine is denoted as arcsine, asin or **sin ^{-1}**.

Suppose a right triangle is taken with side 1, 2, and √3. Now, to calculate angle a, the sine function can be used as-

Now, sine (a) = opposite/hypotenuse i.e ½.

The angle A can now be calculated using the arcsine function.

sin^{−1}(½) = a

This will give the value of angle “a” as 30°

## Sine Identities

Some of the common sine identities are:

- sine (θ) = cos(π/2 − θ) = 1/cosec(θ)
- arcsin (sin θ) = θ, for −π/2 ≤ θ ≤ π/2
- cos
^{2 }(θ) + sin^{2 }(θ) = 1 - Sin (2x) = 2sin (x) cos (x)
- Cos (2x) = cos
^{2}(x) − sin^{2}(x)

There are various other sine identities. The values of other trigonometric functions in terms of sine are also given below which are extremely useful for solving various types of questions.

### Other Trigonometric Functions in Terms of Sine

Trigonometric Functions |
Represented as Sine |

cos θ | \(\pm \sqrt{1\,-\,sin^{2}(\theta)}\) |

tan θ | \(\pm \frac{sin(\theta)}{\sqrt{1\,-\,sin^{2}(\theta)}}\) |

cot θ | \(\pm \frac{\sqrt{1\,-\,sin^{2}(\theta)}}{sin(\theta)}\) |

sec θ | \(\pm \,\frac{1}{\sqrt{1\,-\,sin^{2}(\theta)}}\) |

### Sine Calculus

For sine function **f(x) = sin(x)**,, the derivative and the integral will be given as:

- Derivative of sin(x), f′(x) = cos(x)
- Integral of sin(x), ∫f(x) dx = −cos(x) + C (where C is the constant of integration)

## Law of Sines in Trigonometry

According to law of sines in trigonometry, a relation is established between the sides a, b, and c and angles opposite to those sides A, B and C for an arbitary triangle. The relation is as follows.

In the above diagram, A, B, C are the angles while a, b, c are the lengths of the sides. Now, according to sine law,

\(\frac{a}{Sin\,A}\,=\,\frac{b}{Sin\,B}\,=\,\frac{c}{Sin\,C}\,=\,d\)(Here, d is the diameter of the triangle’s circumcircle).

Check out the law of sines article to know more about it and to know the proof of law of sine, examples, and other details.

**Also Check: **Law of Cosine

### Sine Worksheet (Questions)

- What is the sine of 60?
- What is sin 270?
- What is sin 120 degrees?
- Evaluate sin 360 degrees.
- Derive the value of sin 60 geometrically.
- Find the value of sin 30 degree geometrically and prove sin 30 geometrically.
- Why sin 90 is 1?
- What is sine divided by cosine?
- An aeroplane is flying at 6000 feet above the ground. At what angle should the plane descend to land on the target runway if it is 10,000 feet away from the runway?
- In a triangle ABC, AC is 14 cm, CB is 10 cm, and angle CBA is 63 degrees. Calculate angle CAB and the length of AB.
- For a triangle ABC, sine (ABC) is 0.6 and the length of BC is 12 cm. Find the length of AC (hypotenuse).
- A 55 m rope connects is connected to a point on the ground from the top of a pole. If the rope makes 60 degree angle to the ground, calculate the height of the pole.

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