TRIGONOMETRIC RATIOS OF 180 DEGREE PLUS THETA 

About "Trigonometric ratios of 180 degree plus theta"

Trigonometric ratios of 180 degree plus theta is one of the branches of ASTC formula in trigonometry. 

Trigonometric-ratios of 180 degree plus theta are given below.

sin (180° + θ)  =  - sin θ

cos (180° + θ)  =  - cos θ

tan (180° + θ)  =  tan θ

csc (180° + θ)  =  - csc θ

sec (180° + θ)  =  - sec θ

cot (180° + θ)  =  cot θ

Let us see, how the trigonometric ratios of 180 degree plus theta are determined. 

To know that, first we have to understand ASTC formula. 

The ASTC formula can be remembered easily using the following phrases.

"All Sliver Tea Cups" 

or

"All Students Take Calculus"

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

More clearly 

From the above picture, it is very clear that 

(180° + θ) falls in the third quadrant

In the third quadrant (180° + θ), tan and cot are positive and other trigonometric ratios are negative.

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. 

Evaluation of trigonometric ratios using ASTC formula

Problem 1 :

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 as "cos"

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

Considering the above points, we have 

sin (180° + θ)  =  - sin θ

Let us look at the next stuff on "Trigonometric ratios of 180 degree plus theta"

Problem 2 :

Evaluate : cos (180° + θ)

Solution :

To evaluate cos (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°, "cos" will not be changed as "sin"

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

Considering the above points, we have 

cos (180° + θ)  =  - cos θ

Let us look at the next stuff on "Trigonometric ratios of 180 degree plus theta"

Problem 3 :

Evaluate : tan (180° + θ)

Solution :

To evaluate tan (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°, "tan" will not be changed as "cot"

(iii)  In the III rd quadrant, the sign of "tan" is positive. 

Considering the above points, we have 

tan (180° + θ)  =  tan θ

Let us look at the next stuff on "Trigonometric ratios of 180 degree plus theta"

Problem 4 :

Evaluate : csc (180° + θ)

Solution :

To evaluate csc (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°, "csc" will not be changed as "sec"

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

Considering the above points, we have 

csc (180° + θ)  =   - csc θ

Let us look at the next stuff on "Trigonometric ratios of 180 degree plus theta"

Problem 5 :

Evaluate : sec (180° + θ)

Solution :

To evaluate sec (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°, "sec" will not be changed as "csc"

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

Considering the above points, we have 

sec (180° + θ)  =  - sec θ

Let us look at the next stuff on "Trigonometric ratios of 180 degree plus theta"

Problem 6 :

Evaluate : cot (180° + θ)

Solution :

To evaluate cot (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°, "cot" will not be changed as "tan"

(iii)  In the III rd quadrant, the sign of "cot" is positive. 

Considering the above points, we have 

cot (180° + θ)  =  cot θ

Summary (180 degree plus theta)

sin (180° + θ)  =  - sin θ

cos (180° + θ)  =  - cos θ

tan (180° + θ)  =  tan θ

csc (180° + θ)  =  - csc θ

sec (180° + θ)  =  - sec θ

cot (180° + θ)  =  cot θ

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