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

Signs of Trigonometrical Ratios

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

θ 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.

θ 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. 

θ 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.

θ 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





11 and 12 Grade Math

From Signs of Trigonometrical Ratios to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



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.



Share this page: What’s this?

Recent Articles

  1. 2nd grade math Worksheets | Free Math Worksheets | By Grade and Topic

    Dec 06, 23 01:23 AM

    2nd Grade Math Worksheet
    2nd grade math worksheets is carefully planned and thoughtfully presented on mathematics for the students.

    Read More

  2. Rupees and Paise | Paise Coins | Rupee Coins | Rupee Notes

    Dec 04, 23 02:14 PM

    Different types of Indian Coins
    Money consists of rupees and paise; we require money to purchase things. 100 paise make one rupee. List of paise and rupees in the shape of coins and notes:

    Read More

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

    Dec 04, 23 01:50 PM

    Months of the Year
    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…

    Read More