Which Function Is Undefined When Θ = Π 2 \theta = \frac{\pi}{2} Θ = 2 Π ​ Radians?A. Cos ⁡ Θ \cos \theta Cos Θ B. Cot ⁡ Θ \cot \theta Cot Θ C. Csc ⁡ Θ \csc \theta Csc Θ D. Tan ⁡ Θ \tan \theta Tan Θ

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. One of the key concepts in trigonometry is the definition of trigonometric functions, which are used to describe the relationships between the sides and angles of triangles. In this article, we will discuss which trigonometric function is undefined when $\theta = \frac{\pi}{2}$ radians.

Trigonometric Functions

There are six basic trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. These functions are defined as follows:

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
• Cotangent: $\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$
• Secant: $\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$
• Cosecant: $\csc \theta = \frac{\text{hypotenuse}}{\text{opposite}}$

Undefined Trigonometric Functions

When $\theta = \frac{\pi}{2}$ radians, the trigonometric functions are defined as follows:

• Sine: $\sin \frac{\pi}{2} = 1$
• Cosine: $\cos \frac{\pi}{2} = 0$
• Tangent: $\tan \frac{\pi}{2} = \text{undefined}$
• Cotangent: $\cot \frac{\pi}{2} = 0$
• Secant: $\sec \frac{\pi}{2} = \text{undefined}$
• Cosecant: $\csc \frac{\pi}{2} = \text{undefined}$

As we can see, the tangent, secant, and cosecant functions are undefined when $\theta = \frac{\pi}{2}$ radians.

Why are Tangent, Secant, and Cosecant Undefined?

The tangent, secant, and cosecant functions are undefined when $\theta = \frac{\pi}{2}$ radians because they involve division by zero. In the case of the tangent function, we have:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

When $\theta = \frac{\pi}{2}$ radians, we have $\sin \frac{\pi}{2} = 1$ and $\cos \frac{\pi}{2} = 0$. Therefore, the tangent function is undefined because we are dividing by zero.

Similarly, the secant and cosecant functions are undefined because they involve division by zero.

Conclusion

In conclusion, the tangent, secant, and cosecant functions are undefined when $\theta = \frac{\pi}{2}$ radians. This is because they involve division by zero, which is undefined in mathematics. The sine, cosine, and cotangent functions are defined at this value of $\theta$. It is essential to understand the definition of trigonometric functions and their behavior at different values of $\theta$ to solve problems in trigonometry and other fields.

References

• [1] "Trigonometry" by Michael Corral. Available online at https://math.furman.edu/~mcorral/Trigonometry/Trigonometry.pdf
• [2] "Trigonometric Functions" by Khan Academy. Available online at https://www.khanacademy.org/math/trigonometry

Frequently Asked Questions

• Q: What is the value of $\tan \frac{\pi}{2}$? A: The value of $\tan \frac{\pi}{2}$ is undefined.
• Q: Why is the tangent function undefined at $\theta = \frac{\pi}{2}$ radians? A: The tangent function is undefined at $\theta = \frac{\pi}{2}$ radians because it involves division by zero.
• Q: What is the value of $\sec \frac{\pi}{2}$? A: The value of $\sec \frac{\pi}{2}$ is undefined.
• Q: Why is the secant function undefined at $\theta = \frac{\pi}{2}$ radians? A: The secant function is undefined at $\theta = \frac{\pi}{2}$ radians because it involves division by zero.
• Q: What is the value of $\csc \frac{\pi}{2}$? A: The value of $\csc \frac{\pi}{2}$ is undefined.
• Q: Why is the cosecant function undefined at $\theta = \frac{\pi}{2}$ radians? A: The cosecant function is undefined at $\theta = \frac{\pi}{2}$ radians because it involves division by zero.
  Trigonometric Functions: A Comprehensive Q&A Guide =====================================================

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will provide a comprehensive Q&A guide to trigonometric functions, covering their definitions, properties, and behavior at different values of $\theta$.

Q&A Guide

Q: What is the definition of sine, cosine, and tangent?

A: The sine, cosine, and tangent functions are defined as follows:

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Q: What is the value of $\sin \frac{\pi}{2}$?

A: The value of $\sin \frac{\pi}{2}$ is 1.

Q: What is the value of $\cos \frac{\pi}{2}$?

A: The value of $\cos \frac{\pi}{2}$ is 0.

Q: What is the value of $\tan \frac{\pi}{2}$?

A: The value of $\tan \frac{\pi}{2}$ is undefined.

Q: Why is the tangent function undefined at $\theta = \frac{\pi}{2}$ radians?

A: The tangent function is undefined at $\theta = \frac{\pi}{2}$ radians because it involves division by zero.

Q: What is the definition of cotangent, secant, and cosecant?

A: The cotangent, secant, and cosecant functions are defined as follows:

• Cotangent: $\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$
• Secant: $\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$
• Cosecant: $\csc \theta = \frac{\text{hypotenuse}}{\text{opposite}}$

Q: What is the value of $\cot \frac{\pi}{2}$?

A: The value of $\cot \frac{\pi}{2}$ is 0.

Q: What is the value of $\sec \frac{\pi}{2}$?

A: The value of $\sec \frac{\pi}{2}$ is undefined.

Q: Why is the secant function undefined at $\theta = \frac{\pi}{2}$ radians?

A: The secant function is undefined at $\theta = \frac{\pi}{2}$ radians because it involves division by zero.

Q: What is the value of $\csc \frac{\pi}{2}$?

A: The value of $\csc \frac{\pi}{2}$ is undefined.

Q: Why is the cosecant function undefined at $\theta = \frac{\pi}{2}$ radians?

A: The cosecant function is undefined at $\theta = \frac{\pi}{2}$ radians because it involves division by zero.

Q: What is the range of the sine, cosine, and tangent functions?

A: The range of the sine, cosine, and tangent functions is all real numbers.

Q: What is the range of the cotangent, secant, and cosecant functions?

A: The range of the cotangent, secant, and cosecant functions is all real numbers except 0.

Q: What is the domain of the sine, cosine, and tangent functions?

A: The domain of the sine, cosine, and tangent functions is all real numbers.

Q: What is the domain of the cotangent, secant, and cosecant functions?

A: The domain of the cotangent, secant, and cosecant functions is all real numbers except $\frac{\pi}{2}$ radians.

Q: What is the period of the sine, cosine, and tangent functions?

A: The period of the sine, cosine, and tangent functions is $2\pi$ radians.

Q: What is the period of the cotangent, secant, and cosecant functions?

A: The period of the cotangent, secant, and cosecant functions is $\pi$ radians.

Q: What is the amplitude of the sine, cosine, and tangent functions?

A: The amplitude of the sine, cosine, and tangent functions is 1.

Q: What is the amplitude of the cotangent, secant, and cosecant functions?

A: The amplitude of the cotangent, secant, and cosecant functions is 1.

Q: What is the phase shift of the sine, cosine, and tangent functions?

A: The phase shift of the sine, cosine, and tangent functions is 0.

Q: What is the phase shift of the cotangent, secant, and cosecant functions?

A: The phase shift of the cotangent, secant, and cosecant functions is 0.

Q: What is the vertical shift of the sine, cosine, and tangent functions?

A: The vertical shift of the sine, cosine, and tangent functions is 0.

Q: What is the vertical shift of the cotangent, secant, and cosecant functions?

A: The vertical shift of the cotangent, secant, and cosecant functions is 0.

Q: What is the horizontal shift of the sine, cosine, and tangent functions?

A: The horizontal shift of the sine, cosine, and tangent functions is 0.

Q: What is the horizontal shift of the cotangent, secant, and cosecant functions?

A: The horizontal shift of the cotangent, secant, and cosecant functions is 0.

Q: What is the vertical compression of the sine, cosine, and tangent functions?

A: The vertical compression of the sine, cosine, and tangent functions is 1.

Q: What is the vertical compression of the cotangent, secant, and cosecant functions?

A: The vertical compression of the cotangent, secant, and cosecant functions is 1.

Q: What is the horizontal compression of the sine, cosine, and tangent functions?

A: The horizontal compression of the sine, cosine, and tangent functions is 1.

Q: What is the horizontal compression of the cotangent, secant, and cosecant functions?

A: The horizontal compression of the cotangent, secant, and cosecant functions is 1.

Q: What is the vertical stretch of the sine, cosine, and tangent functions?

A: The vertical stretch of the sine, cosine, and tangent functions is 1.

Q: What is the vertical stretch of the cotangent, secant, and cosecant functions?

A: The vertical stretch of the cotangent, secant, and cosecant functions is 1.

Q: What is the horizontal stretch of the sine, cosine, and tangent functions?

A: The horizontal stretch of the sine, cosine, and tangent functions is 1.

Q: What is the horizontal stretch of the cotangent, secant, and cosecant functions?

A: The horizontal stretch of the cotangent, secant, and cosecant functions is 1.

Q: What is the reflection of the sine, cosine, and tangent functions?

A: The reflection of the sine, cosine, and tangent functions is the same as the original function.

Q: What is the reflection of the cotangent, secant, and cosecant functions?

A: The reflection of the cotangent, secant, and cosecant functions is the same as the original function.

Q: What is the equation of the sine, cosine, and tangent functions?

A: The equation of the sine, cosine, and tangent functions is $y = \sin x$, $y = \cos x$, and $y = \tan x$.

Q: What is the equation of the cotangent, secant, and cosecant functions?

A: The equation of the cotangent, secant, and cosecant functions is $y = \cot x$, $y = \sec x$, and $y = \csc x$.

Q: What is the graph of the sine, cosine, and tangent functions?

A: The graph of the sine, cosine, and tangent functions is a periodic curve that oscillates between positive and negative values.

Q: What is the graph of the cotangent, secant, and cosecant functions?

A: The graph of the cotangent, secant, and cosecant functions is a periodic curve that oscillates between positive and negative values.

Q: What is the amplitude of the sine, cosine, and tangent functions?

A: The amplitude of the sine, cosine, and tangent functions is 1.

Q: What is the amplitude of the cotangent, secant, and cosecant functions?

A: The amplitude of the cotangent, secant, and cosecant functions is 1.

Q: What is the period of the sine, cosine, and tangent functions?

A: The period of the sine, cosine, and tangent functions is $2\pi$ radians.

Q: What is the period of the cotangent, secant, and cosecant functions?

A: The period of the cotangent, secant, and cosecant functions is $\pi$ radians.

Q: What is the phase shift of the sine, cosine, and tangent functions?

A: The phase shift of the sine, cosine, and tangent functions is 0.

Q: What is the phase shift of the