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 (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
11 and 12 Grade Math
From Trigonometrical Ratios of (270° + θ) to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Oct 04, 24 09:47 AM
Oct 04, 24 01:28 AM
Oct 03, 24 03:22 PM
Oct 03, 24 02:22 PM
Oct 03, 24 01:13 PM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.