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Trigonometric Ratios in Quadrants

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°)

trigonometric ratios in quadrants-1sin θ  =  {opposite}/{hypotenuse}~~=~~{y/1};

cos θ  =  {adjacent}/{hypotenuse}~~=~~{x/1}; and

tan θ  =  {opposite}/{adjacent}~~=~~{y/x}.

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

trigonometric ratios in quadrants-2When the point moves to P2 (-x, y) with an angle θ in clockwise direction, point P2 makes an angle of (180° – θ); an obtuse angle in positive direction (anti-clockwise direction).

Using the definition in unit circle and triangle OP2x;

sin (180° – θ)  =  y coordinate of P1  =  y  = sin θ

cos (180° – θ)  =  x coordinate of P1  =  -x  =  -cos θ

tan (180° – θ)  =  {sin (180°~-~theta)}/{cos (180°~-~theta)}~~=~~{sin theta}/{-cos theta}  =  -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°)

trigonometric ratios in quadrants-3Any 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 P3 is the point such that angle XOP3 = 180° + θ.

Hence, by symmetry about the origin, P3 is the point (-x, -y).

sin (180° + θ)  =  y coordinate of P3  =  -y  =  -sin θ

cos (180° + θ)  =  x coordinate of P3  =  -x  =  -cos θ

tan (180° + θ)  =  {sin (180°~+~theta)}/{cos (180°~+~theta)}~~=~~{-sin theta}/{-cos theta}  =  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

trigonometric ratios in quadrants-4Any 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 P4 is the point such that angle XOP4 = 360° – θ)

So by definition

sin (360° – θ)  =  y coordinate of P4  =  -y  =  -sin θ

cos (360° – θ)  =  x coordinate of P4  =  x  =  cos θ

tan (360° – θ)  =  {sin (360°~-~theta)}/{cos (360°~-~theta)}~~=~~{-sin theta}/{cos theta}  =  -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, OP4 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.