Table of Contents

## Trigonometric Functions:

Trigonometric functions are relations between any two of the three sides of a triangle. For the sake of simplicity, a right-angled triangle is taken as a starting point to explain these relations. Among other things, the sides of right-angled triangles are easy to define and grasp. These very ideas will then be extended to all other angles.

**A Right Angle-** Let XOY be any angle Î¸. Take any point P on OY and draw PM perpendicular to OX. A right-angled â–³OMP is formed. If Î¸ is taken as the angle of reference then MP, the side opposite to Î¸ is called the **perpendicular** and OP, the side opposite to the right angle is called the **hypotenuse** and OM, the third side adjacent to the reference angle (other than hypotenuse) is called **adjacent side**.

Now, the three sides OM, OP and MP can be arranged, two at a time in six different ways and hence six ratios can be formed with them. These six ratios are called the trigonometric functions or circular functions or t-ratios and are defined as follows-

(1) The ratio of the perpendicular to the hypotenuse is called the sine of the angle Î¸ and is written as **sin Î¸**.

sin Î¸ = Perpendicular/Hypotenuse = MP/OP = y/r |

(2) The ratio of the adjacent side or base to the hypotenuse is called the cosine of the angle Î¸ and is written as **cos Î¸**.

cos Î¸ = Adjacent Side or Base/Hypotenuse = OM/OP = x/r |

(3) The ratio of the perpendicular to the adjacent sie or base is called the tangent of the angle Î¸ and is written as **tan Î¸**.

tan Î¸ = Perpendicular/Adjacent Side or Base = MP/OM = y/x |

**The following three ratios are reciprocals of the above ratios-**

(1) The ratio of the hypotenuse to the perpendicular is called the cosecant of the angle Î¸ and is written as cosec Î¸.

cosec Î¸ = Hypotenuse/Perpendicular = OP/MP = r/y |

(2) The ratio of the hypotenuse to the adjacent side or base is called the secant of the angle Î¸ and is written as sec Î¸.

sec Î¸ = Hypotenuse/Adjacent Side or Base = OP/OM = r/x |

(3) The ratio of the adjacent side or base to the perpendicular is called the cotangent of the angle Î¸ and is written as cot Î¸.

cot Î¸ = Adjacent Side or Base/Perpendicular = OM/MP =x/y |

## Relationship Between Trigonometric Ratios:

From the above trigonometric functions, it is clear that cosec Î¸ = 1/sin Î¸, sec Î¸ = 1/cos Î¸ and cot Î¸ = 1/tan Î¸. Also, tan Î¸ = sin Î¸/cos Î¸.

Now, in â–³POM, OM^{2} + PM^{2} = OP^{2}â‡’ x ^{2} + y^{2} = r^{2}â‡’ (x/r) ^{2} + (y/r)^{2} = 1â‡’ cos ^{2} Î¸ + sin^{2} Î¸ = 1 …………..(i)Dividing both sides of (i) by cos ^{2} Î¸, we get-1 + tan ^{2} Î¸ = sec^{2} Î¸ …………..(ii)Also, dividing both sides of (i) by sin ^{2} Î¸, one gets-1 + cot ^{2} Î¸ = cosec^{2} Î¸ …………..(iii) |

## Concept of Circular Functions with the Help of Unit Circle:

We consider a circle of the unit radius with its centre at O, the origin of the rectangular coordinate system. We denote the point of intersection of OX and the unit circle by A and take a point P on the circle such that arc AP (measured in anticlockwise direction if Î¸ > 0 and in a clockwise direction if Î¸ < 0) subtends an angle |Î¸| at O. Then, the circular measure of âˆ AOP = arc AP/radius = Î¸/1 = Î¸.

Let the coordinates of P be (x, y) and PM be dropped perpendicular on the X-axis. Then, OM = x and PM = y. The circular functions of Î¸ are defined as sin Î¸ = y, cos Î¸ = x, tan Î¸ = y/x, cosec Î¸ = 1/y, sec Î¸ = 1/x and cot Î¸ = x/y.

Thus, the circular functions of a real number Î¸ and the trigonometric ratios of âˆ AOP are the same.

## Domain and Range of Circular Functions:

From the above figure, it is clear that -1 â‰¤ x â‰¤ 1 and -1 â‰¤ y â‰¤ 1. Again, x > 0, y > 0 in the first quadrant, x < 0, y > 0 in the second quadrant, x < 0, y < 0 in the third quadrant and x > 0, y < 0 in the fourth quadrant.

Thus, in the 1^{st} quadrant, all the six trigonometric ratios are positive. In the 2^{nd} quadrant, x < 0 but y > 0. So, Sin Î¸ = y and cosec Î¸ = 1/y are positive but the remaining four trigonometric ratios are negative.

In the 3^{rd} quadrant, x < 0, y < 0. So, tan Î¸ = y/x and cot Î¸ = x/y are positive but the remaining four t-ratios are negative. Finally, in the 4^{th} quadrant, x > 0, y < 0. So cos Î¸ = x and sec Î¸ = 1/x are positive and the remaining four t-ratios are negative.

The points discussed above may be summarized as follows:

It is clear that sin Î¸ = y and cos Î¸ = x are defined for all real values of Î¸. But tan Î¸ = y/x and sec Î¸ = 1/x are not defined for x = 0. In this case, OP coincides with the Y-axis and so-

Î¸ = Ï€/2, 3Ï€/2, 5Ï€/2, â€¦ or -Ï€/2, -3Ï€/2, -5Ï€/2, â€¦ â‡’ Î¸ = (2n + 1) x/2, where n is any integer. |

Also, cot Î¸ = x/y and cosec Î¸ = 1/y are not defined when y = 0 i.e. when OP coincides with the X-axis. In this case Î¸ = 0, Ï€, 2Ï€, 3Ï€ … or -Ï€, -2Ï€, -3Ï€, … â‡’ Î¸ = nÏ€, where n is any integer.

Again, as -1 â‰¤ x â‰¤ 1 and cos Î¸ = x, we have -1 â‰¤ cos Î¸ â‰¤ 1. Similarly, -1 â‰¤ y â‰¤ 1 and sin Î¸ = y. Therefore, -1 â‰¤ sin Î¸ â‰¤ 1. Clearly, the other four trigonometric ratios may have all real values lying between -âˆž and âˆž.

I | II | III | IV | |
---|---|---|---|---|

sin Î¸ | +ve | +ve | -ve | -ve |

cos Î¸ | +ve | -ve | -ve | +ve |

tan Î¸ | +ve | -ve | +ve | -ve |

cosec Î¸ | +ve | +ve | -ve | -ve |

sec Î¸ | +ve | -ve | -ve | +ve |

cot Î¸ | +ve | -ve | +ve | -ve |

Circular Function | Domain (Î¸ in circular measure) | Range |
---|---|---|

sin | all real numbers | (-1, 1) |

cos | all real numbers | (-1, 1) |

tan | all real numbers other than (2n +1) Ï€/2, where n is any integer | all real values |

cosec | all real numbers other than nÏ€, n being an integer | all real values except -1 and 1 |

sec | all real numbers other than (2n +1) Ï€/2, n being an integer | all real values except -1 and 1 |

cot | all real numbers other than nÏ€, where n is any integer | all real values |

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