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Value of Cot 90°

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Last updated date: 23rd Apr 2024
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Introduction

The value of cot 90 (in degrees) is equivalent to 0 in trigonometry. Trigonometry ratios of complementary angles can be used to generate related formulae based on this value. Many arithmetic issues are solved using the trigonometric ratios table of sin, cos, tan, cosec, sec, and cot for normal angles from 0° to 360°. In trigonometry, we may use several formulae to get the needed values for trigonometric functions. In this post, you will learn what the cot 90 value is and how to calculate the value of cot 90° through derivation.


Cot 90 Value

The Cot 90 value (in degrees) is:


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What are Cot 90 Degrees in Radians?

Cot 90° may be represented as cot π/2 in a circular system. The cot may be represented as cos over sin, as we all know. The unit circle may be used to derive the cot 90 value using this relationship.


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The tangent values of all the degrees may be calculated for a unit circle with a radius of one. We may determine all the trigonometric ratios and values with the assistance of a unit circle drawn on the XY plane.


In the above circle,


cos 90° = 0 and

sin 90° = 1


Now,


cot 90° = cos 90°/sin 90° = 0/1 = 0


Hence, the value of Cot 90 degrees is zero.


Cot 90 Step by Step Derivation

As we all know the value of cot 90 degrees is always zero. Let’s take an overlook how it is calculated,


Recall that cot(θ) = \[\frac{1}{tan(θ)} \]


and the tan(θ) = \[ \frac{sin(θ)}{cos(θ)} \]


Now, we know that,


cot(θ) = \[ \frac{1}{tan(θ)} = \frac{1}{\frac{sin(θ)}{cos(θ)}} = \frac{cos(θ)}{sin(θ)} \]


Now, an instance of θ put 90 degrees,


\[ cot (θ) = \frac{cos(θ)}{sin(θ)} \]


\[ cot (90) = \frac{cos(90)}{sin(90)} \]


By using, the unit circle (below) that,


sin 90° = 1


cos 90° = 0


cot(90) = \[ \frac{0}{1} \]


cot(90) = 0


Hence its proved.


Cot 90 Minus Theta

Let us obtain the formula for cot 90 – theta,


i.e. cot(90° – θ).


cot(90° – θ) = cot(1 × (90° – θ))


Here, 90 degrees is multiplied by an odd number 1. As a result, the trigonometric function will change to its reciprocal form. That indicates we should write "tan" in the following step.


Also, (90° – θ) is in the first quadrant, with a positive tan (all the six functions are positive in this quadrant).


cot(90° – θ) = tan θ


Put θ = y


Then we get,


cot(90° – y) = tan y


And


Again put θ = c


Then we get,


cot(90° – c) = tan c


All of these formulae can be expressed as follows:


cot(π/2 – θ) = tan θ


cot(π/2 – y) = tan y


cot(π/2 – c) = tan c


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Cot 90 Plus Theta

Let us obtain the formula for cot 90 + theta,


i.e. cot(90° + θ).


cot(90° + θ) = cot(1 × (90° + θ))


In this case, 90 degrees is multiplied by an odd number of one. As a result, the trigonometric function will revert to its reciprocal form. That indicates we should write "tan" in the following step.


Also, 90° + θ is located in the second quadrant, where the tan is negative (only sine and cosecant are positive in this quadrant).


Now, cot(90° + θ) = -tan θ


Put θ = y,


Then we get,


cot(90° + y) = -tan y


And,


Put θ = c


cot(90° + c) = -tan c


All of these formulae can be expressed as follows:


cot(π/2 + θ) = -tan θ


cot(π/2 + y) = -tan y


cot(π/2 + c) = -tan c


Where is Cot Undefined

For some particular angles, the value of the cot is undefined. The table of trigonometric ratios below can help you understand how cot is undefined.

Trigonometry Ratios

Angles (In Degrees)

\[0^{\circ} \]

\[30^{\circ} \]

\[45^{\circ} \]

\[60^{\circ} \]

\[90^{\circ} \]

\[1800^{\circ} \]

\[270^{\circ} \]

\[360^{\circ} \]

Angles (In Radians)

\[0^{\circ} \]

\[\frac{\pi}{6}\]

\[\frac{\pi}{4}\]

\[\frac{\pi}{3}\]

\[\frac{\pi}{2}\]

\[\pi\]

\[\frac{3\pi}{2}\]

2\[\pi\]

sin θ

0

\[ \frac{1}{2}\]

\[ \frac{1}{\sqrt{2}}\]

\[ \frac{\sqrt{3}}{2}\]

1

0

-1

0

cos θ

1

\[ \frac{\sqrt{3}}{2}\]

\[ \frac{1}{\sqrt{2}}\]

\[ \frac{1}{2}\]

0

-1

0

1

tan θ

0

\[ \frac{1}{\sqrt{3}}\]

1

\[\sqrt{3}\]

\[\infty\]

0

\[\infty\]

0

cosec θ

\[\infty\]

2

\[\sqrt{2}\]

\[\frac{2}{\sqrt{3}} \]

1

\[\infty\]

-1

\[\infty\]

sec θ

1

\[\frac{2}{\sqrt{3}} \]

\[\sqrt{2}\]

2

\[\infty\]

-1

\[\infty\]

1

cot θ

\[\infty\]

\[\sqrt{3}\]

1

\[\frac{1}{\sqrt{3}} \]

0

\[\infty\]

0

\[\infty\]


Thus, the cot value is undefined (Unknown) for 0°, 180° and 360°. Since the ratio of cos and sin at these angles equals 1/0, these values are represented as undefined or infinite (\[\infty\]).


The value of cot 90 (in degrees) is equivalent to 0 in trigonometry. Many arithmetic issues are solved using the trigonometric Table of sin, cos, tan, cosec, sec, and cot for normal angles from 0° to 360°. This article is more helpful for any student who is confused about the cot values.

Conclusion:

Cot 90 degrees is the cotangent trigonometric function value for a 90-degree angle. Cot 90° has a value of 0.

No, the value of cot 90 has been defined, and it is 0.

The value of cot 90 plus theta is -tan and can be written as $\cot(90^\circ + \theta)$.

As a result, $\cot(90^\circ + \theta) = - \tan \theta$.

The value of cot 90 minus theta is tan and can be written as $\cot(90^\circ - \theta)$. As a result, $\cot(90^\circ - \theta) = \tan \theta$.

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FAQs on Value of Cot 90°

1. What are the 6 basic trigonometric functions?

In trigonometry, there are six functions of an angle that are widely employed.  Their acronyms and names are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

2. What is the Cot of 90?

The cot 90 is equal to 0 and can be expressed as cot 90 degrees, i.e. cot 90°.

3. Why is Cot 90 defined?

Because the ratio of cos 90 degrees to sin 90 degrees has a finite value, cot 90 has a definite value. That is, sin 90° equals 1, cos 90° equals 0, and cot 90° equals cos 90°/sin 90° = 0/1 = 0.

4. What is Cot 90 minus theta?

The value of cot 90 minus theta is tan, and it may be represented as a cot(90° –). As a result, cot(90° –) = tan.