Understanding Cot 90°: Zero or Undefined?
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.
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, $\cot(90^\circ - \theta) = \tan \theta$.
FAQs on What Is the Value of Cot 90°?
1. What is the exact value of cot 90 degrees?
The exact value of cotangent of 90 degrees (cot 90°) is 0. As per the CBSE syllabus for 2025-26, this is a fundamental trigonometric value. In the radian system, this is expressed as cot(π/2) = 0.
2. How is the value of cot 90° calculated using its formula?
The value of cot 90° is calculated using the trigonometric ratio formula: cot(θ) = cos(θ) / sin(θ). To find cot 90°, we use the standard values for cos 90° and sin 90°:
- The value of cos 90° is 0.
- The value of sin 90° is 1.
Substituting these into the formula gives: cot 90° = 0 / 1 = 0.
3. What is the complementary angle identity for cot(90° − θ)?
The key complementary angle identity involving cotangent is cot(90° − θ) = tan(θ). This formula shows the relationship between cotangent and tangent for two angles that add up to 90 degrees, which is a core concept in trigonometry.
4. Why is cot 90° defined and equal to zero, while tan 90° is undefined?
This common point of confusion is explained by their respective formulas:
- For cot 90°: The formula is cos 90° / sin 90°. This becomes 0 / 1, which is a valid mathematical operation that results in 0.
- For tan 90°: The formula is sin 90° / cos 90°. This becomes 1 / 0. Since division by zero is mathematically undefined, tan 90° has no real value.
The crucial difference lies in their denominators; the zero in the numerator for cot 90° makes it zero, while the zero in the denominator for tan 90° makes it undefined.
5. How can the unit circle be used to explain that cot 90° is 0?
The unit circle provides a visual proof. In the unit circle, any point on its circumference has coordinates (cos θ, sin θ). Cotangent is the ratio of these coordinates, cot θ = x / y = cos θ / sin θ.
- At an angle of 90 degrees, the point on the unit circle is at the top, with coordinates (0, 1).
- Here, the x-coordinate (cos 90°) is 0, and the y-coordinate (sin 90°) is 1.
- Calculating the ratio x / y gives 0 / 1, which equals 0.
6. Is it possible to demonstrate the value of cot 90° within a right-angled triangle?
No, the value of cot 90° cannot be demonstrated using the standard SOH-CAH-TOA rules in a right-angled triangle. The trigonometric ratios (like Adjacent/Opposite for cotangent) are defined only for the two acute angles (angles less than 90°) in such a triangle. The value for cot 90° must be determined through other methods like the unit circle or the cos/sin ratio identity, which apply to all angles.
7. What does cot 90° = 0 represent on the graph of the cotangent function?
On the graph of y = cot(x), the value cot 90° = 0 signifies an x-intercept. This is a point where the cotangent curve crosses the x-axis. Specifically, at x = 90° (or π/2 radians), the y-value of the graph is zero. This is different from angles like 0° and 180°, where the graph has vertical asymptotes because the function is undefined.
