Cosec Formula
Trigonometry is the branch of mathematics focused on the relationship between right triangle angles, heights, and distances.
This time, we'll talk about Cosec Cot Formula. Trigonometric ratios are the ratios of the sides of a right triangle. The six main ratios of trigonometry are sin, cos, tan, cot, sec, and cosec. The calculations for both of these equations are different. It makes use of a right-angled triangle's three sides and angles. Let's take a closer look at Cosec Cot Formula.
Cosec Function
The cosecant (cosec) of an angle in a right triangle is the length of the hypotenuse divided by the length of the opposite side.
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In a formula, it is represented as just 'cosec’.
Hence, the formula of cosec is,
Cosec = \[\frac{Hypotenuse}{Opposite side}\]
In addition, the cosec theta formula is a cosec formula, we can write this,
Cosec(theta) = \[\frac{Hypotenuse}{Opposite side}\]
Cosecant, cotangent, and secant are the three trigonometric functions that are rarely used.
Most calculators don't have a button for them, and app feature libraries don't have them too.
Derivations of the more famous three: sin, cos, and tan, can easily be substituted.
The reciprocal of sine can be used to calculate cosecant:
cosec x = \[\frac{1}{sinx}\]
Cot Function
The cotangent of an angle in a right triangle is equal to the length of the opposing side separated by the length of the opposite side.
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In a formula, it is represented as just 'cot'.
Hence cot formula is as follows,
cot = \[\frac{Adjacent}{Opposite side}\]
Cosecant, cotangent, and secant are the three trigonometric functions that are rarely used.
Most calculators don't have a button for them, and app feature libraries don't have them too.
Derivations of the more famous three: sin, cos, and tan, can easily be substituted.
Cotangent (cot) can be derived in two ways:
cot x = \[\frac{1}{tanx}\]
cot x= \[\frac{cosx}{sinx}\]
Relation between Cosec and Cot
It is useful to know the following relations of the cosec cot,
1 + cot2x = cosec2x
d/dxcot x = -cosec2x
d/dxcosec x = -cosec x cot x
So the cosec and cot relation of trigonometric functions are not just related by the Pythagorean identity 1 + cot2x = cosec2x but their derivatives are also dependent on each other.
What Is Cosec Cot Formula?
An acute angle x in a right triangle,
Cosec x is given by,
Cosec x = Hypotenuse / Opposite side
Cot x is given by,
Cot x = Adjacent Side / Opposite Side
The Cosec Cot Formula is given as follows:
1 + cot2 = cosec2
Some Examples and Proof for Understanding
Q.1) Prove that (cosec θ – cot θ)2 = (1 – cos θ)/(1 + cos θ)
Solution:
LHS =(cosec - cot )2
= \[\frac{1}{sin\theta }\]-\[\frac{cos\theta}{sin\theta }\])2
Here, RHS = (1 – cos θ)/(1 + cos θ)
= \[\frac[{1-cos\theta }{sin\theta }]^{2}\]
By rationalising the denominator,
= \[\frac{1-cos\theta }{1+cos\theta }\] \[\frac{1-cos\theta }{1-cos\theta }\]
= \[\frac{(1-cos\theta)^{2} }{1+cos^{2}\theta }\]
= \[\frac{(1-cos\theta)^{2} }{sin^{2}\theta }\]
= \[(\frac{(1-cos\theta) }{sin^{2}\theta })^{2}\]
Therefore, LHS = RHS.
Hence it’s proved.
Q.2) If Sin x = 9/6, find the value of Cosec x?
Cosec x = 1/sinx
= 6/9
Q.3) Find Cot x if Tan P = 4 / 3
Solution: Using Cotangent formula we know that,
Cot x = 1 / Tan x
= 1 / (4 / 3)
= 3 / 4
Thus, Cot x = 3 / 4
Q.4) If cosec x is 4/3 then find the value of sin x.
Answer: Cosec x = 1/ sin x
Sin x = 3/4
FAQs on Cosec cot Formula
1. What is the relation between Pythagoras theorem and trigonometry?
Trigonometry deals with the relationship between the sides and angles of a right-angle triangle, while Pythagoras theorem dealt with the relationship between the three sides of a right-angle triangle.
2. What is the basic relation between sine and cosec?
The cosecant (cosec) is the reciprocal of the sine (sin).
The formula of sine is represented as,
sin x = \[\frac{1}{cosecx}\]
Vice versa,
cosec x =\[\frac{1}{sinx}\]
3. What is the basic relation between tan and cot?
The cotangent (cot) is the reciprocal of the tangent (tan).
The formula of tangent is represented as,
tan x = \[\frac{1}{cotx}\]
Vice versa,
cot x =\[\frac{1}{tanx}\]
