Consider a circle with unit radius and centre O. Let P be a point on the unit circle such that L x OP = θ

We shall find the trigonometric ratios in quadrants one by one, as the ray OP moves on the unit circle in an anticlockwise (positive) direction.

## Angles in the first quadrant (between 0° and 90°)

sin θ = ;

cos θ = ; and

tan θ = .

Here, both x and y are positive; hence, when θ is between 0° and 90°, i.e. the angles in the first quadrant are;

sin θ = + value;

cos θ = + value; and

tan θ = + value.

## Angles in the second quadrant (between 90° and 180°) i.e., 180° – θ

When the point moves to P_{2} (-x, y) with an angle θ in clockwise direction, point P_{2} makes an angle of (180° – θ); an obtuse angle in positive direction (anti-clockwise direction).

Using the definition in unit circle and triangle OP_{2}x;

sin (180° – θ) = y coordinate of P_{1}_{ }= y = sin θ

cos (180° – θ) = x coordinate of P_{1} = -x = -cos θ

tan (180° – θ) = = -tan θ.

Here θ is the angle in anti-clockwise direction using symmetry.

In the second quadrant, when angles are obtuse (θ between 90° and 180°);

sin θ = + value;

cos θ = – value; and

tan θ = – value.

## Angles in the third quadrant (180° + θ°) or (270° – θ°)

Any angle in the third quadrant can be related to an acute angle θ. P(x, y) makes and angle θ = L x OP with x-axis and P_{3} is the point such that angle XOP_{3} = 180° + θ.

Hence, by symmetry about the origin, P_{3} is the point (-x, -y).

sin (180° + θ) = y coordinate of P_{3} = -y = -sin θ

cos (180° + θ) = x coordinate of P_{3} = -x = -cos θ

tan (180° + θ) = = tan θ.

Hence the angles in the third quadrant lie between 180° and 270°

sin θ = – value;

cos θ = – value; and

tan θ = + value.

## Angles in the fourth quadrant (between 90° and 180°) i.e. 360° – θ

Any angle in the fourth quadrant can be related to an acute angle θ. P (x, y) is the point in the first quadrant such that angle XOP = θ, and P_{4} is the point such that angle XOP_{4} = 360° – θ)

So by definition

sin (360° – θ) = y coordinate of P_{4} = -y = -sin θ

cos (360° – θ) = x coordinate of P_{4} = x = cos θ

tan (360° – θ) = = -tan θ.

Here the angles in the fourth quadrant have the following values of its trigonometric ratios:

sin θ = – value;

cos θ = + value; and

tan θ = – value.

Note that in the fourth quadrant, OP_{4} is the terminal ray of a negative angle -θ.

sin (360° – θ) = sin (-θ) = -sin θ

cos (360° – θ) = cos (-θ) = cos θ

tan (360° – θ) = -tan θ = tan(-θ)

We now look at some examples of calculating trigonometric values in quadrants.