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The cosecant function is a trigonometric function and is the reciprocal of the sine function. It is abbreviated as csc x or cosec x, where x is the angle.
- In a right-angled triangle, the cosecant is the ratio of the hypotenuse to the perpendicular.
- Since it is the reciprocal of sine, we can write the cosecant formula as cosec x = 1 / sin x.
- Trigonometry is a branch of mathematics that studies the relationship between the angles and sides of a right-angled triangle.
- It's used to calculate the unknown sides of a right triangle, as well as the angles that form between them.
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Key Terms: Cosecant formula, Hypotenuse, Cosex x, csc x, Sine function, Cosecant function, Trigonometry, Pythogoras theorem, Triangle, Angle
What is Cosecant?
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Cosecant is derived from the sine ratio. It is abbreviated as ‘cosec’ or ‘csc’ and has a period of 2π, which is similar to sine and cosine.
- The cosecant function is the reciprocal of the sine function.
- Thus, the cosecant function becomes undefined whenever the sine function is equal to zero (0).
Trigonometry ratios are defined by representing the relationship between sides and angles of a right-angled triangle. There are six ratios which are the core of trigonometry. These ratios are:
- Sine
- Cosine
- Tangent
- Cotangent
- Secant
- Cosecant
Out of these six trigonometry ratios, sine, cosine, and tangent are basic while the other three, secant, cosecant, and cotangent are derived. The cosecant function is the complement of the secant function.
Cosecant Formula
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A right-angled triangle has three sides namely hypotenuse, perpendicular (the opposite side), and base (the adjacent side).
When taking an angle as a reference
- The largest side is the hypotenuse
- The side opposite to the angle is the perpendicular, and
- The side where both hypotenuse and opposite rest is the base.
The length of the hypotenuse, when divided by the length of the perpendicular, gives the cosecant of the angle in a right-angled triangle. Therefore, the cosecant formula of an angle is given by
| \(Cosec\: x = \frac{Hypotense}{Perpendicular}\) |
Also, the cosecant is the reciprocal of the sine value. Thus,
| \(Cosec\:x = \frac{1}{sin\:x}\) |
Cosecant Formula
Cosecant Ratios Table
The cosecant ratio table for various standard angles with their respective value is given below:
| Angle | Value |
|---|---|
| cosec 0° | Undefined |
| cosec 30° | 2 |
| cosec 45° | √2 |
| cosec 60° | 2/√3 |
| cosec 90° | 1 |
| cosec 180° | Undefined |
Cosecant Function in Quadrants
The cosecant function has different signs in different quadrants which are described below:
| Degree | Quadrant | Sign of Secant Function |
|---|---|---|
| 0° to 90° | First | positive |
| 90° to 180° | Second | positive |
| 180° to 270° | Third | negative |
| 270° to 360° | Fourth | negative |
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Solved Examples of Cosecant
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| Example: Calculate Cosec x if Sin x = \(\frac{5}{9}\) Solution: As Cosec x = 1/ Sin x = 1/(\(\frac{5}{9}\)) = 9/5 So, Cosec x = 9/5 Example: Find cosec x if cot x = 3/4 Solution: As we know, cosec2x - cot2x = 1 ⇒ cosec2x = 1 + (3/4)2 = 25/16 Thus, cosec x = 5/4. Example: Find cosec x if sin x = 4/7 Solution: Since we know, cosec x = 1/sin x Thus, cosec x = 1/(4/7) = \(\frac{7}{4}\) Example: Find the height of a right-angled triangle whose hypotenuse is 14 units and base angle is 30°. Solution: Given that, θ = 60° and H = 14 units and let perpendicular be P units Using the cosecant formula, cosec θ = H/P cosec 30° =14/P 2 = 14/P P = 14/2 P = 7 Therefore, the height of the right-angle triangle is 7 Units. |
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Things to Remember
- The cosecant of an angle in a right triangle is calculated by dividing the length of the hypotenuse by the length of the opposite side
- The cosecant function is the reciprocal of the sine function.
- It becomes undefined whenever the sine function is equal to zero (0).
- The cosecant formula is given by cosec x = (hypotenuse / perpendicular) = 1 / sin x.
- Cosec has a period of 2\(\pi\), which is similar to sine and cosine.
- The cosecant function is positive in the first and second quadrants.
Sample Questions
Ques. If cos x = 12/13, find the value of cosec x using the formula. (2 Marks)
Ans. We have, cos x = 12/13.
So we get, sin x = 5/13.
Using the formula we get,
cosec x = 1/sin x
= 1/(5/13)
= 13/5
Ques. If tan x = 12/5, find the value of cosec x using the formula. (2 Marks)
Ans. We have, tan x = 12/5.
So we get, sin x = 12/13 and cos x = 5/13.
Using the formula we get,
cosec x = 1/sin x
= 1/(12/13)
= 13/12
Ques. If sin x = 3/5, find the value of cosec x using the formula. (2 Marks)
Ans. We have, sin x = 3/5.
Using the formula we get,
cosec x = 1/sin x
= 1/(3/5)
= 5/3
Ques. If cot x = 15/8, find the value of cosec x using the formula. (2 Marks)
Ans. We have, cot x = 15/8.
So we get, cos x = 15/17 and sin x = 8/17.
Using the formula we get,
cosec x = 1/sin x
= 1/(8/17)
= 17/8
Ques. If sec x = 5/3, find the value of cosec x using the formula. (2 Marks)
Ans. We have, sec x = 5/3.
So we get, cos x = 3/5 and sin x = 4/5.
Using the formula we get,
cosec x = 1/sin x
= 1/(4/5)
= 5/4
Ques. Represent cosec in terms of cos function. (2 Marks)
Ans. As we know, cosecθ = 1/sinθ
And sinθ = √1 - cos2θ
Thus, cosecθ = 1/(√1 - cos2θ)
Ques. What will be the value of cosec 270°? (2 Marks)
Ans. Since, sin 270° = -1
Thus, cosec 270° = 1/-1 = -1
Ques. Find the values of the cosecant of angles A and C of triangle right angled at B, if AB = 12, AC = 13. (2 Marks)
Ans. We know that cosec x = Hypotenuse / Perpendicular.
Let us evaluate the value of BC first.
AC² = AB² + BC²
BC = √(AC² - AB²)
= √(13² - 12²)
= √(169 - 144)
= √25
= 5 units
Ques. What is the difference between secant and cosecant? (2 Marks)
Ans. The secant function is the reciprocal of the cosine function and the Cosecant function is the reciprocal of the sine function. Secant is the ratio of hypotenuse and adjacent side whereas cosecant is the ratio of the Hypotenuse and Opposite Side.
Ques. What is the period of cosec function? (2 Marks)
Ans. The values of the cosecant function repeat after every 2π radians, so the period of cosec x is equal to 2π radians (360 degrees).
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