# 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 $$\overrightarrow{OA}$$ 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 $$\overrightarrow{OA}$$ 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 $$\overrightarrow{OA}$$ 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 $$\overrightarrow{OA}$$ 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 $$\overrightarrow{OA}$$.

Trigonometric Functions