TRIGONOMETRY QUADRANT FORMULAS

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

Evaluate the following trig values

Problem 1 :

cos 120

Solution :

cos 120 lies in 2^nd quadrant.

Reference angle in 2^nd quadrant will be, 180 - θ

cos 120 = cos (180 - 120)

= -cos 60

Considering ASTC, in 2^nd quadrant sin θ and cosec θ will be positive.

= -1/2

Then the value of cos 120 is -1/2.

Problem 2 :

cot 240

Solution :

cot 240 lies in 3^rd quadrant.

Reference angle in 3^rd quadrant will be, θ - 180

Considering ASTC, in 3^rd quadrant tan θ and cot θ will be positive.

cot 240 = cot (240 - 180)

= cot 60

√2/2

cot 60 = 1/tan 60

= 1/√3

Then the value of cot 240 is 1/√3.

Problem 3 :

sin (-330)

Solution :

sin (-330) = - sin 330

sin 330 lies in 4^th quadrant.

Reference angle in 4^th quadrant will be, 360 - θ

Considering ASTC, in 4^th quadrant cos θ and sec θ will be positive.

sin 330 = -sin (360 - 330)

= -sin 30

= -1/2

Then the value of sin (-330) is -1/2.

Problem 4 :

cos 135°

Solution :

135° lies in 2^nd quadrant.

Reference angle in 2^nd quadrant will be, 180 - θ

Considering ASTC, in 2^nd quadrant sin θ and cosec θ will be positive.

sin 135 = sin (180 - 135)

= sin 45

= √2/2

Then the value of sin 135 is √2/2.

Problem 5 :

cos 300°

Solution :

300° lies in 4^th quadrant.

Reference angle in 4^th quadrant will be, 360 - θ

Considering ASTC, in 4^th quadrant cos θ and sec θ will be positive.

cos 300 = cos (360 - 300)

= cos 60

= 1/2

Then the value of cos 300 is 1/2.

Problem 6 :

cot 315°

Solution :

315° lies in 4^th quadrant.

Reference angle in 4^th quadrant will be, 360 - θ

Considering ASTC, in 4^th quadrant cos θ and sec θ will be positive.

cot 315 = cot (360 - 315)

= -cot 45

= -1

Then the value of cot 315 is -1.

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