- Position of a point P in the Cartesian plane with respect to co-ordinate axes is represented by the ordered pair (x, y).
- Trigonometry is the science of relationships between the sides and angles of a right-angled triangle.
- Trigonometric Ratios: Ratios of sides of right triangle are called trigonometric ratios.

Consider triangle ABC right-angled at B. These ratios are always defined with respect to acute angle ‘A’ or angle ‘C. - If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ratios of an angle can be easily determined.
- How to identify sides: Identify the angle with respect to which the t-ratios have to be calculated. Sides are always labelled with respect to the ‘θ’ being considered.

Let us look at both cases:

In a right triangle ABC, right-angled at B. Once we have identified the sides, we can define six t-Ratios with respect to the sides.

case I | case II |

(i) sine A = | (i) sine C = |

(ii) cosine A = | (ii) cosine C = |

(iii) tangent A = | (iii) tangent C = |

(iv) cosecant A = | (iv) cosecant C = |

(v) secant A = | (v) secant C = |

(v) cotangent A = | (v) cotangent C = |

Note from above six relationships:

cosecant A = , secant A = , cotangent A = ,

However, it is very tedious to write full forms of t-ratios, therefore the abbreviated notations are:

sine A is sin A

cosine A is cos A

tangent A is tan A

cosecant A is cosec A

secant A is sec A

cotangent A is cot A

**TRIGONOMETRIC IDENTITIES**

An equation involving trigonometric ratio of angle(s) is called a trigonometric identity, if it is true for all values of the angles involved. These are:

tan θ =

cot θ =

- sin² θ + cos² θ = 1 ⇒ sin² θ = 1 – cos² θ ⇒ cos² θ = 1 – sin² θ
- cosec² θ – cot² θ = 1 ⇒ cosec² θ = 1 + cot² θ ⇒ cot² θ = cosec² θ – 1
- sec² θ – tan² θ = 1 ⇒ sec² θ = 1 + tan² θ ⇒ tan² θ = sec² θ – 1
- sin θ cosec θ = 1 ⇒ cos θ sec θ = 1 ⇒ tan θ cot θ = 1

**ALERT:**

A t-ratio only depends upon the angle ‘θ’ and stays the same for same angle of different sized right triangles.

**Value of t-ratios of specified angles:**

∠A | 0° | 30° | 45° | 60° | 90° |

sin A | 0 | 1 | |||

cos A | 1 | 0 | |||

tan A | 0 | 1 | √3 | not defined | |

cosec A | not defined | 2 | √2 | 1 | |

sec A | 1 | √2 | 2 | not defined | |

cot A | not defined | √3 | 1 | 0 |

The value of sin θ and cos θ can never exceed 1 (one) as opposite side is 1. Adjacent side can never be greater than hypotenuse since hypotenuse is the longest side in a right-angled ∆.

**‘t-RATIOS’ OF COMPLEMENTARY ANGLES**

If ∆ABC is a right-angled triangle, right-angled at B, then

∠A + ∠C = 90° [∵ ∠A + ∠B + ∠C = 180° angle-sum-property]

or ∠C = (90° – ∠A)

**Thus, ∠A and ∠C are known as complementary angles and are related by the following relationships:**

sin (90° -A) = cos A; cosec (90° – A) = sec A

cos (90° – A) = sin A; sec (90° – A) = cosec A

tan (90° – A) = cot A; cot (90° – A) = tan A