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.


11 and 12 Grade Math

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