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

• Basic Trigonometric Ratios and Their Names
• Restrictions of Trigonometrical Ratios
• Reciprocal Relations of Trigonometric Ratios
• Quotient Relations of Trigonometric Ratios
  
• Limit of Trigonometric Ratios
  
• Trigonometrical Identity
• Problems on Trigonometric Identities
• Elimination of Trigonometric Ratios
• Eliminate Theta between the equations
• Problems on Eliminate Theta
• Trig Ratio Problems
• Proving Trigonometric Ratios
• Trig Ratios Proving Problems
• Verify Trigonometric Identities
• Trigonometrical Ratios of 0°
• Trigonometrical Ratios of 30°
• Trigonometrical Ratios of 45°
• Trigonometrical Ratios of 60°
• Trigonometrical Ratios of 90°
• Trigonometrical Ratios Table
• Problems on Trigonometric Ratio of Standard Angle
  
• Trigonometrical Ratios of Complementary Angles
• Rules of Trigonometric Signs
• Signs of Trigonometrical Ratios
  
• All Sin Tan Cos Rule
• Trigonometrical Ratios of (- θ)
• Trigonometrical Ratios of (90° + θ)
• Trigonometrical Ratios of (90° - θ)
• Trigonometrical Ratios of (180° + θ)
• Trigonometrical Ratios of (180° - θ)
• Trigonometrical Ratios of (270° + θ)
• Trigonometrical Ratios of (270° - θ)
• Trigonometrical Ratios of (360° + θ)
• Trigonometrical Ratios of (360° - θ)
• Trigonometrical Ratios of any Angle
• Trigonometrical Ratios of some Particular Angles
• Trigonometric Ratios of an Angle
• Trigonometric Functions of any Angles
• Problems on Trigonometric Ratios of an Angle
• Problems on Signs of Trigonometrical Ratios

11 and 12 Grade Math

From Rules of Trigonometric Signs to HOME PAGE