Cosec theta (cosec θ) is a trigonometric function defined as the reciprocal of the sine function (sin θ).
According to the reference provided:
Cosec ϴ is a trigonometric function and is the reciprocal of Sin ϴ, ( 1/ sin ϴ)........ Cosec ϴ = hypotenuse/perpendicular or h/p.18-Jan-2017
Here's a breakdown:
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Reciprocal Relationship:
- cosec θ = 1 / sin θ
- This means that if you know the value of sin θ, you can easily find cosec θ by taking its reciprocal.
-
Ratio in a Right-Angled Triangle:
- cosec θ is also defined as the ratio of the hypotenuse to the perpendicular (opposite) side in a right-angled triangle.
- cosec θ = hypotenuse / perpendicular (h/p)
Practical Understanding
To illustrate further, consider a right-angled triangle:
| Side | Abbreviation |
|---|---|
| Hypotenuse | h |
| Perpendicular | p |
| Base | b |
Based on the definitions, we can define the following relationships:
- sin θ = perpendicular / hypotenuse = p/h
- cosec θ = 1/sin θ = hypotenuse / perpendicular = h/p
Table of Key Trigonometric Functions and their Reciprocals
| Function | Abbreviation | Definition | Reciprocal | Reciprocal Abbreviation |
|---|---|---|---|---|
| Sine | sin θ | perpendicular / hypotenuse = p/h | Cosecant | cosec θ |
| Cosine | cos θ | base/ hypotenuse=b/h | Secant | sec θ |
| Tangent | tan θ | perpendicular / base = p/b | Cotangent | cot θ |
Example:
- If sin θ is 1/2, then cosec θ would be 2/1, or simply 2.
- If the hypotenuse of a right-angled triangle is 5 and the perpendicular is 3, then cosec θ would be 5/3.
