# Signs of Trigonometrical Ratios

Here we will discuss about the signs of trigonometrical ratios.

Let a rotating line $$\overrightarrow{OA}$$ rotates about O in the anti-clockwise direction or clockwise direction. Suppose starting from the rotating line $$\overrightarrow{OA}$$ as the initial position $$\overrightarrow{OX}$$ take ∠XOA = θ. Take a point B on $$\overrightarrow{OA}$$ and a line is drawn which is $$\overline{BC}$$ perpendicular to $$\overrightarrow{OA}$$ (or $$\overrightarrow{OX'}$$). Therefore, by the definition of trigonometrical ratios of the angle θ of the right-angled triangle OBC are:

 sin θ = CB/OB = opposite side/hypotenuse; cos θ = OC/OB = adjacent side/hypotenuse; tan θ = CB/OC = opposite side/adjacent side; csc θ = OB/CB = hypotenuse/opposite side sec θ = OB/OC = hypotenuse/adjacent side; cot θ = OC/CB = adjacent side/opposite side

According to the value of θ the final arm $$\overrightarrow{OA}$$ would be in the first quadrant or second quadrant or third quadrant or fourth quadrant:

Case 1: When the final arm $$\overrightarrow{OA}$$ lies in the first quadrant

According to the trigonometric rules, we get

OC is positive,

CB is positive and

OB is positive.

Therefore, according to the definitions of trigonometrical ratios the values of all trigonometrical ratios i.e. sin θ, cos θ, tan θ, csc θ, sec θ and cot θ are positive.

Case 2: When the final arm $$\overrightarrow{OA}$$ lies in the second quadrant.

According to the trigonometric rules, we get

OC is negative,

CB is positive and

OB is positive.

Therefore, according to the definitions of trigonometrical ratios the values of sin θ and csc θ are positive and the other trigonometrical ratios i.e. cos θ, tan θ, sec θ and cot θ are negative.

Case 3:  When the final arm $$\overrightarrow{OA}$$ lies in the third quadrant.

According to the trigonometric rules, we get

OC is negative;

CB is negative and

OB is positive.

Therefore, according to the definitions of trigonometrical ratios the values of tan θ and cot Ѳ are positive and the other trigonometrical ratios i.e. sin θ, cos θ, sec θ and csc θ are negative.

Case 4: When the final arm $$\overrightarrow{OA}$$ lies in the fourth quadrant.

According to the trigonometric rules, we get

OC is positive;

CB is negative and

OB is positive.

Therefore, according to the definitions of trigonometrical ratios the values of cos θ and sec θ are positive and the other trigonometrical ratios i.e. sin θ, tan θ, csc θ and cot θ are negative.

Trigonometric Functions

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Relation between Diameter Radius and Circumference |Problems |Examples

Apr 22, 24 05:19 PM

Relation between diameter radius and circumference are discussed here. Relation between Diameter and Radius: What is the relation between diameter and radius? Solution: Diameter of a circle is twice

2. ### Circle Math | Terms Related to the Circle | Symbol of Circle O | Math

Apr 22, 24 01:35 PM

In circle math the terms related to the circle are discussed here. A circle is such a closed curve whose every point is equidistant from a fixed point called its centre. The symbol of circle is O. We…

3. ### Preschool Math Activities | Colorful Preschool Worksheets | Lesson

Apr 21, 24 10:57 AM

Preschool math activities are designed to help the preschoolers to recognize the numbers and the beginning of counting. We believe that young children learn through play and from engaging

4. ### Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

Apr 20, 24 05:39 PM

There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…