The Cos 60 value is 0.5. Cos 60 or Cosine 60 is the ratio of the Base (Adjacent) and Hypotenuse in a right-angle triangle with the angle between the base and hypotenuse equalling 60 degrees (or *π*/3, in radians) as shown below. This article discusses the Cos 60 value and its proof.

## Cos 60 degree value

As mentioned above, the Value of Cos 60 is 0.5 (or ½ in fractions). The value of the Cosine of an acute angle is evaluated in the context of a right-angled triangle. Cosine is evaluated by taking out the ratio of the base of a right-angled triangle (also known as its adjacent) and its hypotenuse. The value of Cos moves from 1 to 0 when the angle moves from 0 to 90°.

### Cos 60 degrees value proof

The invention of the Sine and Cosines of angles can be traced back to ancient Indian Astronomy during the Gupta Period. Early Indian Astronomers used trigonometric functions such as *Jyā, Koṭi-jyā *and* Utkrama-jyā* to evaluate certain values in the arc of a circle. These, however, were not the functions of angles as in modern trigonometry. Jya can be roughly translated to Perpendicular and Koṭi-jyā to Base of modern trigonometry.

In a typical right-angled triangle with angles of 30, 60 and 90 degrees (as shown in the figure below), the ratio of sides is

Perpendicular : Base : Hypotenuse = √3 : 1 : 2

This ratio satisfies the Pythagoras Theorem as shown below

(Perpendicular)^{2} + (Base)^{2} = (Hypotenuse)^{2}

It can also be written as

(Opposite)^{2} + (Adjacent)^{2} = (Hypotenuse)^{2}

(√3)^{2} + (1)^{2} = (2)^{2}

3 + 1 = 4

4 = 4

LHS = RHS

Hence, if we evaluate Cos 60 from it, we get the following:

Cos θ= Adjacent / Hypotenuse

Cos 60 = 1/2

### The Trigonometric Table

Cos 60 value can also be found in the following table.

0° (0) | 30° (π/6) | 45° (π/4) | 60° (π/3) | 90° (π/2) | |

Sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |

Cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |

Tan | 0 | 1/√3 | 1 | √3 | Not Defined |

Cot | Not Defined | √3 | 1 | 1/√3 | 0 |

Sec | 1 | 2/√3 | √2 | 2 | Not Defined |

Cosec | Not Defined | 2 | √2 | 2/√3 | 1 |

Also Read:

The value of a function can be evaluated from the following table

FUNCTION | RATIO |

Sin θ | Opposite / Hypotenuse |

Cos θ | Adjacent / Hypotenuse |

Tan θ | Opposite / Adjacent |

Cot θ | Adjacent / Opposite |

Sec θ | Hypotenuse / Adjacent |

Cosec θ | Hypotenuse / Opposite |

The Cos 60 value can be calculated from the reciprocal also. The values of functions are the reciprocal of some other function. The following table presents you with the functions and their reciprocals.

FUNCTION | RECIPROCAL |

Sin θ | Cosec θ |

Cos θ | Sec θ |

Tan θ | Cot θ |

Cot θ | Tan θ |

Sec θ | Cos θ |

Cosec θ | Sin θ |

#### Evaluating Cos 60 Value

The value of Cos 60 can be evaluated from the following methods as well

Cos θ = Sin (90 – θ)

Hence

Cos 60 = Sin (90 – 60)

Cos 60 = Sin 30

We can get the value of Sin 30 from The Trigonometric Table above

Sin 30 = ½

Hence Cos 60 = ½

#### Cos 60 value using Reciprocals

Sec θ = 1/ Cos θ

Hence, Cos θ = 1/ Sec θ

Cos 60 = 1/ Sec 60

From The Trigonometric Table above, Sec 60 = 2

Hence,

Cos 60 = ½

#### Evaluating the value of Cos θ from Tan θ and Sin θ

Tan θ = Sin θ / Cos θ

Cos θ = Sin θ / Tan θ

If we want to find the value of Cos 60

Cos 60 = Sin 60 / Tan 60

Cos 60 = √3 / 2 ÷ √3

Hence, we get

Cos 60 = 1 / 2

#### Evaluating Cos θ with the help of Sin^{2}θ

Using the formula

Sin^{2}θ + Cos^{2}θ = 1

To find the Cos 60 Value, we substitute θ with 60

Sin^{2}60 + Cos^{2}60 = 1

As we know that, Sin 60 = √3/2

Hence, Sin^{2}60 = ¾

Cos^{2}60 = 1 – Sin^{2}60

Cos^{2}60 = 1 – ¾

Cos^{2}60 = ¼

Cos 60 = √(¼)

Cos 60 = ½