TRIGONOMETRY QUADRANT FORMULAS

In this section, you will learn how the values of six trigonometric change in different quadrants.

To understand how the values of trigonometric ratios change in different quadrants, first we have to understand ASTC rule. 

ASTC Rule :

ASTC rule is nothing but "all sin tan cos" rule in trigonometry. 

The "all sin tan cos" rule can be remembered easily using the following phrases.

"All Sliver Tea Cups" 

or

"All Students Take Calculus"

ASTC rule has been explained clearly in the figure given below.

Important Conversions :

When we have the angles 90° and 270° in the trigonometric ratios in the form of

(90° + θ)

(90° - θ)

(270° + θ)

(270° - θ)

We have to do the following conversions, 

sin θ <------> cos θ

tan θ <------> cot θ

csc θ <------> sec θ

For example,

sin (270° + θ)  =  - cos θ

cos (90° - θ)  =  sin θ

For the angles 0° or 360° and  180°, we should not make the above conversions. 

Division of Quadrants 

(90° - θ) -------> I st Quadrant

(90° + θ) and (180° - θ) -------> II nd Quadrant

(180° + θ) and (270° - θ) -------> III rd Quadrant

(270° + θ), (360° - θ) and (- θ) -------> IV th Quadrant

we can evaluate the following trigonometric ratios. 

270° + θ

sin (270°+θ)  =  -cos θ

cos (270°+θ)  =  sin θ

tan (270°+θ)  =  -cot θ

csc (270°+θ)  =  -sec θ

sec (270°+θ)  =  cos θ

cot (270°+θ)  =  -tan θ

Angles Greater than or Equal to 360°

If the angle is equal to or greater than 360°, we have to divide the given angle by 360 and take the remainder. 

For example,  

(i) Let us consider the angle 450°.

When we divide 450° by 360, we get the remainder 90°. 

Therefore, 450°  =  90°

(ii) Let us consider the angle 360°

When we divide 360° by 360, we get the remainder 0°.

Therefore, 360°  =  0°

Based on the above two examples, we can evaluate the following trigonometric ratios. 

sin (360° - θ)  =  sin (0° - θ)  =  sin (- θ)  =  - sin θ

cos (360° - θ)  =  cos (0° - θ)  =  cos (- θ)  =  cos θ

tan (360° - θ)  =  tan (0° - θ)  =  tan (- θ)  =  - tan θ

csc (360° - θ)  =  csc (0° - θ)  =  csc (- θ)  =  - csc θ

sec (360° - θ)  =  sec (0° - θ)  =  sec (- θ)  =  sec θ

cot (360° - θ)  =  cot (0° - θ)  =  cot (- θ)  =  - cot θ

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