SIGN OF TRIGONOMETRIC FUNCTION

To find the sign of s trigonometric function, we should know  about the four quadrants. 

• The angle which lies between 0 and 90° is known as first quadrant. 
• The angle which lies between 90° and 180° is known as second quadrant. 
• The angle which lies between 180° and 270° is known as third quadrant. 
• The angle which lies between 270° and 360° is known as fourth quadrant. 

More clearly, 

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

(90° + θ) (or) (90° - θ)

(270° + θ) (or) (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. 

Examples

Example 1 :

Evaluate : cos (270° - θ)

Solution :

To evaluate cos (270° - θ), we have to consider the following important points. 

(i)  (270° - θ) will fall in the III rd quadrant. 

(ii)  When we have 270°, "cos" will become "sin"

(iii)  In the III rd quadrant, the sign of "cos" is negative. 

Considering the above points, we have 

cos (270° - θ)  =  - sin θ

Example 2 :

Evaluate : sin (180° + θ)

Solution :

To evaluate sin (180° + θ), we have to consider the following important points. 

(i)  (180° + θ) will fall in the III rd quadrant. 

(ii)  When we have 180°, "sin" will not be changed

(iii)  In the III rd quadrant, the sign of "sin" is negative. 

Considering the above points, we have 

sin (180° + θ)  =  - sin θ

Based on the above two examples, 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

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