Sin, Cos and Tan

January 26, 2021January 9, 2019 by Sanja Dodos

Contents hide

2 i) First Quadrant:

3 ii) Second Quadrant:

4 iii) Third Quadrant:

5 iv) Fourth Quadrant:

6 Graphs of sin, cos and tan

Summary

$sin\theta \quad =\quad \frac { opposite }{ hypotenuse }$
$cos\theta \quad =\quad \frac { adjacent }{ hypotenuse }$
$tan\theta \quad =\quad \frac { opposite }{ adjacent }$
First Quadrant: All are positive in this quadrant.
Second Quadrant: Only sin is positive in this quadrant.
Third Quadrant: Only tan is positive in this quadrant.
Fourth Quadrant: Only cos is positive in this quadrant.

$sin\theta \quad =\quad \frac { opposite }{ hypotenuse }$

$cos\theta \quad =\quad \frac { adjacent }{ hypotenuse }$ $tan\theta \quad =\quad \frac { opposite }{ adjacent }$

We now consider angles in cartesian plane. We divide the plane into four quadrants in the anticlockwise sense as shown in the diagram. An angle formed, when the line OA rotates anticlockwise is taken to be positive, while clockwise rotation gives negative angles.

The angle $\alpha$ is always acute regardless of the value of $\theta$ and is called the ”associated acute angle”.

i) First Quadrant:

In the first quadrant, $\theta$ is acute and all values of x, y and r are positive, therefore all the trigonometric ratios are positive.

$cos\theta \quad =\quad \frac { x }{ r } \quad >\quad 0,$

$cos\theta \quad =\quad \frac { x }{ r } \quad >\quad 0,$

$tan\theta \quad =\quad \frac { y }{ x } \quad >\quad 0$

ii) Second Quadrant:

Acute angle $\alpha \quad =\quad 180\quad -\quad \theta$

Hence:

$sin\alpha \quad =\quad \frac { +y }{ r } \quad >\quad 0,\quad (+)$

$cos\alpha \quad =\quad \frac { -x }{ r } \quad <\quad 0,\quad (-)$

$tan\alpha \quad =\quad \frac { y }{ -x } \quad <\quad 0,\quad (-)$

iii) Third Quadrant:

In this quadrant angle $\alpha$ and $\theta$ are related by $\theta \quad =\quad 180\quad +\quad \alpha$

Hence:

$sin\alpha \quad =\quad \frac { -y }{ r } \quad <\quad 0,\quad (-)$

$cos\alpha \quad =\quad \frac { -x }{ r } \quad <\quad 0,\quad (-)$

$tan\alpha \quad =\quad \frac { -y }{ -x } \quad >\quad 0,\quad (+)$

iv) Fourth Quadrant:

In this quadrant angle $\alpha$ and $\theta$ are related by $\theta \quad =\quad 360\quad -\quad \alpha$

Hence:

$sin\alpha \quad =\quad \frac { -y }{ r } \quad <\quad 0,\quad (-)$

$cos\alpha \quad =\quad \frac { x }{ r } \quad >\quad 0,\quad (+)$

$tan\alpha \quad =\quad \frac { -y }{ x } \quad <\quad 0,\quad (-)$

The signs of the trigonometric ratios in the four quadrants can be summarised in the following “ACTS” diagram.

Graphs of sin, cos and tan

sin $\theta$

cos $\theta$

tan $\theta$

Reference