Rules of Trigonometric Signs

In this section we will learn about the rules of trigonometric signs. On a plane paper let O be a fixed point. Draw two mutually perpendicular lines \(\overrightarrow{XOX'}\) and \(\overrightarrow{YOY'}\) through O divide the plane paper into four quadrants.

We know that, the distance measured from O along \(\overrightarrow{XO}\) is positive and that along \(\overrightarrow{OX'}\) is negative; similarly again, the distance from O along \(\overrightarrow{OY}\) is positive and that along \(\overrightarrow{OY'}\) is negative.

Now, take a rotating line \(\overrightarrow{OA}\) rotates about O in the clockwise or anti-clockwise direction and starting from the initial position angle ∠XOA = θ. Depending on the value of θ the final arm \(\overrightarrow{OA}\) may be in the first quadrant or second quadrant or third quadrant or fourth quadrant. Take a point B on \(\overrightarrow{OA}\) and draw \(\overline{BC}\) perpendicular to \(\overrightarrow{OX}\) (or, \(\overrightarrow{OX'}\)).

Diagram 1:

(i) \(\overline{OC}\) will be positive if it is measured from O along \(\overrightarrow{OX}\)

(ii) \(\overline{CB}\) will be positive if it is measured from O along \(\overrightarrow{OY}\)

(iii) \(\overline{OB}\) is positive of the final arm \(\overrightarrow{OA}\)

Rules of Trigonometric Signs

Diagram 1


Diagram 2:

(i) \(\overline{OC}\) will be negative if it is measured from O along \(\overrightarrow{OX'}\)

(ii) \(\overline{CB}\) will be positive if it is measured from O along \(\overrightarrow{OY}\)

(iii) \(\overline{OB}\) is positive of the final arm \(\overrightarrow{OA}\)

Rules of  Signs

Diagram 2


Diagram 3:

(i) \(\overline{OC}\) will be negative if it is measured from O along \(\overrightarrow{OX'}\)

(ii) \(\overline{CB}\) will be negative if it is measured from O along \(\overrightarrow{OY'}\)

(iii) \(\overline{OB}\) is positive of the final arm \(\overrightarrow{OA}\)

Rules of  Trig Signs

Diagram 3


Diagram 4:

(i) \(\overline{OC}\) will be positive if it is measured from O along \(\overrightarrow{OX}\)

(ii) \(\overline{CB}\) will be negative if it is measured from O along \(\overrightarrow{OY'}\)

(iii) \(\overline{OB}\) is positive of the final arm \(\overrightarrow{OA}\)

Rules of Trigonometric Sign

Diagram 4

Therefore, the rules of trigonometric signs of the sides of the right-angled triangle OBC are as follows:

(i) \(\overline{OC}\) will be positive if it is measured from O along \(\overrightarrow{OX}\) as shown in the diagram 1 and diagram 4

(ii) \(\overline{OC}\) will be negative if it is measured from O along \(\overrightarrow{OX'}\) as shown in the diagram 2 and diagram 3

(iii) \(\overline{CB}\) will be positive if it is measured from O along \(\overrightarrow{OY}\) as shown in the diagram 1 and diagram 2

(iv) \(\overline{CB}\) will be negative if it is measured from O along \(\overrightarrow{OY'}\) as shown in the diagram 3 and diagram 4

(v) \(\overline{OB}\) is positive for all positions of the final arm \(\overrightarrow{OA}\).


11 and 12 Grade Math

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