Here we will discuss about the signs of trigonometrical ratios.
Let a rotating line \(\overrightarrow{OA}\) rotates about O in the anticlockwise 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 rightangled 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.
11 and 12 Grade Math
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