Trigonometrical Ratios of (180° - θ)

What are the relations among all the trigonometrical ratios of (180° - θ)?

In trigonometrical ratios of angles (180° - 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 (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

Using the above proved results we will prove all six trigonometrical ratios of (180° - θ).

sin (180° - θ) = sin (90° + 90° - θ)

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

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

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

 

cos (180° - θ) = cos (90° + 90° - θ)

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

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

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

 

tan (180° - θ) = cos (90° + 90° - θ)

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

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

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


csc (180° - θ) = \(\frac{1}{sin (180° - \Theta)}\)

                    = \(\frac{1}{sin  \Theta}\), [since sin (180° - θ) = sin θ]

Therefore, csc (180° - θ) = csc θ;


sec (180° - θ) = \(\frac{1}{cos (180° - \Theta)}\)

                    = \(\frac{1}{- cos  \Theta}\), [since cos (180° - θ) = - cos θ]

Therefore, sec (180° - θ) = - sec θ

and

cot (180° - θ) = \(\frac{1}{tan (180° - \Theta)}\)

                    = \(\frac{1}{- tan  \Theta}\), [since tan (180° - θ) = - tan θ]

Therefore, cot (180° - θ) =  - cot θ.


Solved examples:

1. Find the value of sec 150°.

Solution:

sec 150° = sec (180 - 30)°

            = - sec 30°; since we know, sec (180° - θ) = - sec θ

            = - \(\frac{2}{√3}\)


2. Find the value of tan 120°.

Solution:

tan 120° = tan (180 - 60)°

            = - tan 60°; since we know, tan (180° - θ) = - tan θ

            = - √3





11 and 12 Grade Math

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