## 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 θ. Take any point P on OY and draw PM perpendicular to OX. A right-angled △OMP is formed. If θ is taken as the angle of reference then MP, the side opposite to θ 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 θ and is written as **sin θ**.

sin θ = 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 θ and is written as **cos θ**.

cos θ = 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 θ and is written as **tan θ**.

tan θ = 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 θ and is written as cosec θ.

cosec θ = 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 θ and is written as sec θ.

sec θ = 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 θ and is written as cot θ.

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

## Relationship Between Trigonometric Ratios:

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

Now, in △POM, OM^{2} + PM^{2} = OP^{2}⇒ x ^{2} + y^{2} = r^{2}⇒ (x/r) ^{2} + (y/r)^{2} = 1⇒ cos ^{2} θ + sin^{2} θ = 1 …………..(i)Dividing both sides of (i) by cos ^{2} θ, we get-1 + tan ^{2} θ = sec^{2} θ …………..(ii)Also, dividing both sides of (i) by sin ^{2} θ, one gets-1 + cot ^{2} θ = cosec^{2} θ …………..(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 θ > 0 and in a clockwise direction if θ < 0) subtends an angle |θ| at O. Then, the circular measure of ∠AOP = arc AP/radius = θ/1 = θ.

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 θ are defined as sin θ = y, cos θ = x, tan θ = y/x, cosec θ = 1/y, sec θ = 1/x and cot θ = x/y.

Thus, the circular functions of a real number θ and the trigonometric ratios of ∠AOP are the same.

## Domain and Range of Circular Functions:

From the above figure, it is clear that -1 ≤ x ≤ 1 and -1 ≤ y ≤ 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 θ = y and cosec θ = 1/y are positive but the remaining four trigonometric ratios are negative.

In the 3^{rd} quadrant, x < 0, y < 0. So, tan θ = y/x and cot θ = x/y are positive but the remaining four t-ratios are negative. Finally, in the 4^{th} quadrant, x > 0, y < 0. So cos θ = x and sec θ = 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 θ = y and cos θ = x are defined for all real values of θ. But tan θ = y/x and sec θ = 1/x are not defined for x = 0. In this case, OP coincides with the Y-axis and so-

θ = π/2, 3π/2, 5π/2, … or -π/2, -3π/2, -5π/2, … ⇒ θ = (2n + 1) x/2, where n is any integer. |

Also, cot θ = x/y and cosec θ = 1/y are not defined when y = 0 i.e. when OP coincides with the X-axis. In this case θ = 0, π, 2π, 3π … or -π, -2π, -3π, … ⇒ θ = nπ, where n is any integer.

Again, as -1 ≤ x ≤ 1 and cos θ = x, we have -1 ≤ cos θ ≤ 1. Similarly, -1 ≤ y ≤ 1 and sin θ = y. Therefore, -1 ≤ sin θ ≤ 1. Clearly, the other four trigonometric ratios may have all real values lying between -∞ and ∞.

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

sin θ | +ve | +ve | -ve | -ve |

cos θ | +ve | -ve | -ve | +ve |

tan θ | +ve | -ve | +ve | -ve |

cosec θ | +ve | +ve | -ve | -ve |

sec θ | +ve | -ve | -ve | +ve |

cot θ | +ve | -ve | +ve | -ve |

Circular Function | Domain (θ in circular measure) | Range |
---|---|---|

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

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

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

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

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

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

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