# Trigonometric Ratios Re-Defined

Consider a unit circle with centre O and a revolving line OP. Let OP revolve in an anticlockwise direction and arrive at a point (*a, b*) in the 1

^{st}quadrant (

*a >*0

*and*

*b*> 0).

PM is drawn perpendicular to OA Trigonometric ratios (circular functions) are defined as follows:

We notice all the ratios are positive, since

*a*and

*b*are positive.

**Note:-**From the definition, observe that tan Î¸ and sec Î¸ and not defined if

*a*= 0 and cot Î¸ and cosec Î¸ are not defined if

*b*=0.

Observation: not defined.

Move to II quadrant now The angle rotated through is Ï€ âˆ’ Î¸ .

PM =

*b*and OM = âˆ’

*a*

**Definitions**:-

Here we notice that sin and cosec functions remain the same and the other ratios change sign. (because negative in the second quadrant)

**Observation:-**When

*b*= 0, angle is Ï€

so we notice:

*sin*Ï€ = 0,

*cos*Ï€ = âˆ’ 1,

*tan*Ï€ = 0,

*cosec*Ï€ ,

*cot*Ï€ not defined

and

*sec*Ï€ = âˆ’ 1.

**Table II**

â€‹Signs of Trigonometry Ratios

â€‹Signs of Trigonometry Ratios

Quadrant |
sinÎ¸ |
cosÎ¸ |
tanÎ¸ |
cosecÎ¸ |
secÎ¸ |
cotÎ¸ |

I |
+ |
+ |
+ |
+ |
+ |
+ |

II |
+ |
- |
- |
+ |
- |
- |

III |
- |
- |
+ |
- |
- |
+ |

IV |
- |
+ |
- |
- |
+ |
- |

The same idea can be briefly represented like this

**Table III**

Values of Trigonometric Ratios for 0 âˆ’ 2Ï€

Values of Trigonometric Ratios for 0 âˆ’ 2Ï€

0 | ||||||||

0 | 1 | 0 | -1 | 0 | ||||

1 | 0 | -1 | 0 | 1 | ||||

0 | 1 | 0 | 0 |

**Note:-**Any positive number divided by 0 is considered to be âˆž and negative number divided by 0 is -âˆž.

**Important Results**

- For all values of Î¸ , cos(âˆ’ Î¸ ) = cosÎ¸ and

This is because (âˆ’ Î¸ ) is an angle in the 4

^{th}quadrant (if Î¸ < 90Â°) then cosine is positive, sine and tangent are negative.

This property leads to the conclusion:

**Cosine is an even function**

**Sine is an odd function**

**Tangent is an odd function**

The reciprocals of sin, cos and tan have the same property as the original functions

- Values of sin and
*cos*functions repeat after one complete revolution. So we say that sin and*cos*have a period of 2Ï€

The Period of tan function is Ï€ .

Eg:

Values of sin and cos functions fluctuate between -1 and +1 where as tan values change from -âˆž to âˆž

**Table IV Values of T - Ratios in the four quadrants**

Quadrant |
sin x | cos x | tan x | cosec x | sec x | cot x |

I |
0 to 1 | 1 to 0 | 0 to âˆž | âˆž to 1 | 1 to âˆž | âˆž to 0 |

II |
1 to 0 | 0 to -1 | -âˆž to 0 | 1 to âˆž | -âˆž to -1 | 0 to -âˆž |

III |
0 to -1 | -1 to 0 | 0 to âˆž | -âˆž to -1 | -1 to -âˆž | âˆž to 0 |

IV |
-1 to 0 | 0 to 1 | -âˆž to 0 | -1 to -âˆž | âˆž to 1 | 0 to -âˆž |

**Rule:-**

- Write the angle in the form
- Determine the quadrant in which the terminal side of the angle lies.
- Determine the sign of the given Trigonometric ratio in that particular quadrant (Table II)
- If k is even, T - ratio of the given angle is the same as what is given in the question.
- If
*k*is odd, change the function to its complementary ratio.

Table V,

*Î¸*is acute

**Example 1:**

Use the rules given above to find sin 2460Â°

**Example 2:**

1. Evaluate

**Solution:-**

**Example 3:**

Find the value of other trigonometric ratios if

**Solution:-**

[II quadrant â†’ only sin

*x*& cosec

*x*are positive]

[III quadrant â†’ only tan

*x*& cot

*x*are positive]

**Example 4:**

If,evaluate.

**Solution:**

**Example 5:**

**Solution:**