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
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 example:
1. Find the value of sin 225°.
Solution:
sin (225)° = sin (180 + 45)°
= - sin 45°; since we know sin (180° + θ) = - sin θ
= - \(\frac{1}{√2}\)
2. Find the value of sec 210°.
Solution:
sec (210)° = sec (180 + 30)°
= - sec 30°; since we know sec (180° + θ) = - sec θ
= - \(\frac{1}{√2}\)
3. Find the value of tan 240°.
Solution:
tan (240)° = tan (180 + 60)°
= tan 60°; since we know tan (180° + θ) = tan θ
= √3
● Trigonometric Functions
11 and 12 Grade Math
From Trigonometrical Ratios of (180° + 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.
Jun 13, 24 02:51 AM
Jun 13, 24 02:28 AM
Jun 13, 24 12:11 AM
Jun 12, 24 01:11 PM
Jun 11, 24 07:15 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.