Signs of the Trigonometric Ratios

The signs of the trigonometric ratios depend on the quadrant in which the terminal side of the angle lies. We always take the length of the terminal side OP = r to be positive.


Thus, the sign of $sin \Theta = \frac{y}{r}$ depends on 'y' , $cos \Theta = \frac{x}{r}$ depends on 'x' and $tan \Theta = \frac{y}{x}$ depends on x and y both. Similarly the signs of other trigonometric ratios are decided by the signs of 'x' and /or 'y'.

Signs of the trigonometric ratios according to the quadrants

In first quadrant (I quadrant):
In the I-quadrant, both the coordinates x and y are positive.
x > 0 , y > 0
∴ $sin \Theta = \frac{y}{r}$ > 0 , $cos \Theta = \frac{x}{r}$ > 0, $tan \Theta = \frac{y}{x}$ > 0

$csc \Theta = \frac{r}{y}$ > 0 , $sec \Theta = \frac{r}{x}$ > 0, $cot \Theta = \frac{x}{y}$ > 0
Thus in the first quadrant all trigonometric functions are positive.

In second quadrant (II quadrant):
In the II-quadrant, the coordinates x is negative and y is positive.
x < 0 , y > 0
∴ $sin \Theta = \frac{y}{r}$ > 0 , $cos \Theta = \frac{x}{r}$ < 0, $tan \Theta = \frac{y}{x}$ < 0

$csc \Theta = \frac{r}{y}$ > 0 , $sec \Theta = \frac{r}{x}$ < 0, $cot \Theta = \frac{x}{y}$ < 0
Thus in the second quadrant sine and cosecant ratios are positive all other ratios are negative.

In third quadrant (III quadrant):
In the III-quadrant, the coordinates x and y both are negative.
x < 0 , y < 0
∴ $sin \Theta = \frac{y}{r}$ < 0 , $cos \Theta = \frac{x}{r}$ < 0, $tan \Theta = \frac{y}{x}$ > 0

$csc \Theta = \frac{r}{y}$ < 0 , $sec \Theta = \frac{r}{x}$ < 0, $cot \Theta = \frac{x}{y}$ > 0
Thus in the third quadrant all trigonometric ratios are negative except tangent and cotangent.

In fourth quadrant (IV quadrant):
In the III-quadrant, the coordinates x is positive and y is negative.
x > 0 , y < 0
∴ $sin \Theta = \frac{y}{r}$ < 0 , $cos \Theta = \frac{x}{r}$ > 0, $tan \Theta = \frac{y}{x}$ < 0

$csc \Theta = \frac{r}{y}$ < 0 , $sec \Theta = \frac{r}{x}$ > 0, $cot \Theta = \frac{x}{y}$ < 0
Thus in the fourth quadrant all trigonometric ratios are negative except cosine and secant.



11th grade math

From Signs of the trigonometric ratios to Home