Trigonometrical Ratios of (270° - θ)

What are the relations among all the trigonometrical ratios of (270° - θ)?

In trigonometrical ratios of angles (270° - θ) we will find the relation between all six trigonometrical ratios.

We know that,

sin (90° - θ) = cos θ

cos (90° - θ) = sin θ

tan (90° - θ) = cot θ

csc (90° - θ) = sec θ

sec (90° - θ) = csc θ

cot (90° - θ) = tan θ

and

 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

Using the above proved results we will prove all six trigonometrical ratios of (270° - θ).

sin (270° - θ) = sin [180° + 90° - θ]

                   = sin [180° + (90° - θ)]          

                   = - sin (90° - θ), [since sin (180° + θ) = - sin θ]

Therefore, sin (270° - θ) = - cos θ, [since sin (90° - θ) = cos θ]

 

cos (270° - θ) = cos [180° + 90° - θ]

                    = cos [180° + (90° - θ)]

                    = - cos (90° - θ), [since cos (180° + θ) = - cos θ]

Therefore, cos (270° - θ) = - sin θ, [since cos (90° - θ) =  sin θ]

 

tan (270° - θ) = tan [180° + 90° - θ]

                    = tan [180° + (90° - θ)]

                    = tan (90° - θ), [since tan (180° + θ) = tan θ]

Therefore, tan (270° - θ) = cot θ, [since tan (90° - θ) = cot θ]

 

csc (270° - θ) = \(\frac{1}{sin (270° - \Theta)}\)

                    = \(\frac{1}{- cos \Theta}\), [since sin (270° - θ) = - cos θ]

Therefore, csc (270° - θ) = - sec θ;

 

sec (270° - θ) = \(\frac{1}{cos (270° - \Theta)}\)

                    = \(\frac{1}{- sin \Theta}\), [since cos (270° - θ) = -sin θ]

Therefore, sec (270° - θ) = - csc θ

and

cot (270° - θ) = \(\frac{1}{tan (270° - \Theta)}\)

                    = \(\frac{1}{cot \Theta}\), [since tan (270° - θ) = cot θ]

Therefore, cot (270° - θ) = tan θ.


Solved examples:

1. Find the value of cot 210°.

Solution:

cot 210° = cot (270 - 60)°

            = tan 60°; since we know, cot (270° - θ) = tan θ

            = √3


2. Find the value of cos 240°.

Solution:

cos 240° = cos (270 - 30)°

            = - sin 30°; since we know, cos (270° - θ) = - sin θ

            = - 1/2





11 and 12 Grade Math

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