## Trigonometrical Ratios or Functions

In the right angled triangle OMP, we have base = OM = x, perpendicular =PM = y and hypotenues = OP =r. We define the following trigonometric ratio which are also known as trigonometric function.

**(1) Relation between trigonometric ratios (functions)**

- sin θ . cosec θ = 1
- tan θ . cot θ = 1
- cos θ . sec θ = 1
- \(\tan { \theta } =\quad \frac { \sin { \theta } }{ \cos { \theta } } \)
- \(\cot { \theta } =\quad \frac { \cos { \theta } }{ \sin { \theta } } \)

**(2) Fundamental trigonometric identities**

- sin
^{2}θ + cos^{2}θ = 1 - 1 + tan
^{2}θ = sec^{2}θ - 1 + cot
^{2}θ = cosec^{2}θ

**(3) Sign of trigonometrical ratios or functions**

Their signs depends on the quadrant in which the terminal side of the angle lies.

**In brief:** A crude aid to memorise the signs of trigonometrical ratio in different quadrant. “Add Sugar To Coffee”.

**Algorithm:** First determine the sign of the trigonometric function.

If θ is measured from X*‘*OX i.e., {(π ± θ, 2π – θ)} then retain the original name of the function.

If θ is measured from Y*‘*OY i.e., {π/2 ± θ, 3π/2 ± θ}, then change sine to cosine, cosine to sine, tangent to cotangent, cot to tan, sec to cosec and cosec to sec.

**(4) Variations in values of trigonometric functions in different quadrants:**

Let X*‘*OX and Y*‘*OY be the coordinate axes. Draw a circle with centre at origin O and radius unity.

Let M(x, y) be a point on the circle such that then ∠AOM = θ then x = cos θ and y = sin θ; −1 ≤ cos θ ≤ 1 and −1 ≤ cos θ ≤ 1 for all values of θ.

II-Quadrant (S) | I-Quadrant (A) |

sin θ ⟶ decreases from 1 to 0 | sin θ ⟶ increases from 0 to 1 |

cos θ ⟶ decreases from 0 to – 1 | cos θ ⟶ decreases from 1 to 0 |

tan θ ⟶ increases from – ∞ to 0 | tan θ ⟶ increases from 0 to ∞ |

cot θ ⟶ decreases from 0 to – ∞ | cot θ ⟶ decreases from ∞ to 0 |

sec θ ⟶ increases from – ∞ to – 1 | sec θ ⟶ increases from 1 to ∞ |

cosec θ ⟶ increases from 1 to ∞ | cosec θ ⟶ decreases from ∞ to 1 |

III-Quadrant (T) | IV-Quadrant (C) |

sin θ ⟶ decreases from 0 to – 1 | sin θ ⟶ increases from – 1 to 0 |

cos θ ⟶ increases from – 1 to 0 | cos θ ⟶ increases from 0 to 1 |

tan θ ⟶ increases from 0 to ∞ | tan θ ⟶ increases from – ∞ to 0 |

cot θ ⟶ decreases from ∞ to 0 | cot θ ⟶ decreases from 0 to – ∞ |

sec θ ⟶ decreases from – 1 to – ∞ | sec θ ⟶ decreases from ∞ to 1 |

cosec θ ⟶ increases from – ∞ to – 1 | cosec θ ⟶ decreases from –1 to –∞ |

### Trigonometrical ratios of allied angles

Two angles are said to be allied when their sum or difference is either zero or a multiple of 90°.

### Trigonometrical ratios for various angles

θ | 0 | ?/6 | ?/4 | ?/3 | ?/2 | ? | 3?/2 | 2? |

sin θ | 0 | 1/2 | 1/√2 | 3/√2 | 1 | 0 | –1 | 0 |

cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 | –1 | 0 | 1 |

tan θ | 0 | 1/√3 | 1 | √3 | ∞ | 0 | ∞ | 0 |

### Trigonometrical ratios for some special angles

### Trigonometrical ratios in terms of each other

### Formulae for the trigonometric ratios of sum and differences of two angles

### Formulae for the trigonometric ratios of sum and differences of three angles

(1) *sin(A + B + C) = sin A cos B cos C + cos A sin B cos C + cos A cos B sin C – sin A sin B sin C*

*or sin(A + B + C) = cos A cos B cos C(tan A + tan B + tan C – tan A. tan B. tan C)*

(2) *cos(A + B + C) = cos A cos B cos C – sin A sin B cos C + sin A cos B sin C – cos A sin B sin C*

*cos(A + B + C) = cos A cos B cos C(1 – tan A tan B – tan B tan C – tan C tan A)*

(5) *sin(A _{1} + A_{2} + …… + A_{n}) = cos A_{1} cos A_{2} ….. cos A_{n}(S_{1} – S_{3} + S_{5} – S_{7} + ….)*

(6)

*cos(A*

_{1}+ A_{2}+ …… + A_{n}) = cos A_{1}cos A_{2}….. cos A_{n}(1 – S_{2}+ S_{4}– S_{6}….)where,

*S*= The sum of the tangents of the separate angles.

_{1}= tan A_{1}+ tan A_{2}+ ….. + tan A_{n}*S*= The sum of the tangents taken two at a time.

_{2}= tan A_{1}tan A_{2}+ tan A_{1}tan A_{3}+ ….*S*= Sum of tangents three at a time, and so on.

_{3}= tan A_{1}tan A_{2}tan A_{3}+ tan A_{2}tan A_{3}tan A_{4}+ ….If

*A*, then

_{1}= A_{2}= …. = A_{n}= A*S*

_{1}= n tan A, S_{2}=^{n}C_{2}tan^{2}A, S_{3}=^{n}C_{3}tan^{3}A,……(8)

*sin nA = cos*

^{n}A(^{n}C_{1}tan A –^{n}C_{3}tan^{3}A +^{n}C_{5}tan^{5}A – …..)(9)

*cos nA = cos*

^{n}A(1 –^{n}C_{2}tan^{2}A +^{n}C_{4}tan^{4}A – …..)(11)

*sin nA + cos nA = cos*

^{n}A(1 +^{n}C_{1}tan A –^{n}C_{2}tan^{2}A –^{n}C_{3}tan^{3}A +^{n}C_{4}tan^{4}A +^{n}C_{5}tan^{5}A –^{n}C_{6}tan^{6}A –…..)(12)

*sin nA – cos nA = cos*

^{n}A(–1 +^{n}C_{1}tan A) +^{n}C_{2}tan^{2}A –^{n}C_{3}tan^{3}A –^{n}C_{4}tan^{4}A +^{n}C_{5}tan^{5}A +^{n}C_{6}tan^{6}A + …..)### Formulae to transform the product into sum or difference

Therefore, we find out the formulae to transform the sum or difference into product.