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.

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

sin (270° + θ) = sin [1800 + 90° + θ]

                   = sin [1800 + (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 [1800 + 90° + θ]

                    = cos [I 800 + (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 [1800 + 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 csc 315°.

Solution:

csc 315° = sec (270 + 45)°

             = - sec 45°; since we know, csc (270° + θ) = - sec θ

             = - √2

2. Find the value of cos 330°.

Solution:

cos 330° = cos (270 + 60)°

             = sin 60°; since we know, cos (270° + θ) = sin θ

             = $\frac{√3}{2}$

● Trigonometric Functions

• Basic Trigonometric Ratios and Their Names
• Restrictions of Trigonometrical Ratios
• Reciprocal Relations of Trigonometric Ratios
• Quotient Relations of Trigonometric Ratios
  
• Limit of Trigonometric Ratios
  
• Trigonometrical Identity
• Problems on Trigonometric Identities
• Elimination of Trigonometric Ratios
• Eliminate Theta between the equations
• Problems on Eliminate Theta
• Trig Ratio Problems
• Proving Trigonometric Ratios
• Trig Ratios Proving Problems
• Verify Trigonometric Identities
• Trigonometrical Ratios of 0°
• Trigonometrical Ratios of 30°
• Trigonometrical Ratios of 45°
• Trigonometrical Ratios of 60°
• Trigonometrical Ratios of 90°
• Trigonometrical Ratios Table
• Problems on Trigonometric Ratio of Standard Angle
  
• Trigonometrical Ratios of Complementary Angles
• Rules of Trigonometric Signs
• Signs of Trigonometrical Ratios
  
• All Sin Tan Cos Rule
• Trigonometrical Ratios of (- θ)
• Trigonometrical Ratios of (90° + θ)
• Trigonometrical Ratios of (90° - θ)
• Trigonometrical Ratios of (180° + θ)
• Trigonometrical Ratios of (180° - θ)
• Trigonometrical Ratios of (270° + θ)
• Trigonometrical Ratios of (270° - θ)
• Trigonometrical Ratios of (360° + θ)
• Trigonometrical Ratios of (360° - θ)
• Trigonometrical Ratios of any Angle
• Trigonometrical Ratios of some Particular Angles
• Trigonometric Ratios of an Angle
• Trigonometric Functions of any Angles
• Problems on Trigonometric Ratios of an Angle
• Problems on Signs of Trigonometrical Ratios

11 and 12 Grade Math

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