Evaluate The Expression: $\[ \cot^2 \frac{\pi}{6} + \operatorname{cosec} \frac{5\pi}{6} + 3 \tan^2 \frac{\pi}{6} = 6 \\]

Introduction

In this article, we will evaluate the given trigonometric expression involving cotangent, cosecant, and tangent functions. The expression is ${ \cot^2 \frac{\pi}{6} + \operatorname{cosec} \frac{5\pi}{6} + 3 \tan^2 \frac{\pi}{6} = 6 }$. We will use the properties and identities of trigonometric functions to simplify and evaluate the expression.

Understanding the Trigonometric Functions

Before we proceed with the evaluation, let's recall the definitions of the trigonometric functions involved:

• Cotangent (cot): The cotangent of an angle $\theta$ is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle. It is the reciprocal of the tangent function, i.e., $\cot \theta = \frac{1}{\tan \theta}$.
• Cosecant (cosec): The cosecant of an angle $\theta$ is defined as the ratio of the hypotenuse to the opposite side in a right-angled triangle. It is the reciprocal of the sine function, i.e., $\cosec \theta = \frac{1}{\sin \theta}$.
• Tangent (tan): The tangent of an angle $\theta$ is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.

Evaluating the Expression

Now, let's evaluate the given expression:

$${ \cot^2 \frac{\pi}{6} + \operatorname{cosec} \frac{5\pi}{6} + 3 \tan^2 \frac{\pi}{6} = 6 }$$

We can start by evaluating each term separately:

• $\cot^2 \frac{\pi}{6}$: Since $\cot \frac{\pi}{6} = \frac{1}{\tan \frac{\pi}{6}} = \frac{1}{\sqrt{3}}$, we have $\cot^2 \frac{\pi}{6} = \left( \frac{1}{\sqrt{3}} \right)^2 = \frac{1}{3}$.
• $\operatorname{cosec} \frac{5\pi}{6}$: Since $\sin \frac{5\pi}{6} = \frac{1}{2}$, we have $\operatorname{cosec} \frac{5\pi}{6} = \frac{1}{\sin \frac{5\pi}{6}} = \frac{1}{\frac{1}{2}} = 2$.
• $3 \tan^2 \frac{\pi}{6}$: Since $\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$, we have $\tan^2 \frac{\pi}{6} = \left( \frac{1}{\sqrt{3}} \right)^2 = \frac{1}{3}$. Therefore, $3 \tan^2 \frac{\pi}{6} = 3 \cdot \frac{1}{3} = 1$.

Simplifying the Expression

Now that we have evaluated each term, we can substitute the values back into the original expression:

$${ \cot^2 \frac{\pi}{6} + \operatorname{cosec} \frac{5\pi}{6} + 3 \tan^2 \frac{\pi}{6} = 6 }$$

$${ \frac{1}{3} + 2 + 1 = 6 }$$

$${ \frac{1}{3} + 3 = 6 }$$

$${ \frac{10}{3} = 6 }$$

Conclusion

The given expression ${ \cot^2 \frac{\pi}{6} + \operatorname{cosec} \frac{5\pi}{6} + 3 \tan^2 \frac{\pi}{6} = 6 }$ is not true. The correct value of the expression is $\frac{10}{3}$, which is not equal to 6.

Final Answer

The final answer is $\boxed{\frac{10}{3}}$.

Introduction

In our previous article, we evaluated the given trigonometric expression involving cotangent, cosecant, and tangent functions. The expression is ${ \cot^2 \frac{\pi}{6} + \operatorname{cosec} \frac{5\pi}{6} + 3 \tan^2 \frac{\pi}{6} = 6 }$. We found that the expression is not true, and the correct value is $\frac{10}{3}$.

Q&A

Here are some frequently asked questions related to the evaluation of the expression:

Q: What is the value of $\cot^2 \frac{\pi}{6}$?

A: The value of $\cot^2 \frac{\pi}{6}$ is $\frac{1}{3}$.

Q: What is the value of $\operatorname{cosec} \frac{5\pi}{6}$?

A: The value of $\operatorname{cosec} \frac{5\pi}{6}$ is 2.

Q: What is the value of $3 \tan^2 \frac{\pi}{6}$?

A: The value of $3 \tan^2 \frac{\pi}{6}$ is 1.

Q: What is the final answer to the expression ${ \cot^2 \frac{\pi}{6} + \operatorname{cosec} \frac{5\pi}{6} + 3 \tan^2 \frac{\pi}{6} = 6 }$?

A: The final answer to the expression is $\boxed{\frac{10}{3}}$.

Q: Why is the expression ${ \cot^2 \frac{\pi}{6} + \operatorname{cosec} \frac{5\pi}{6} + 3 \tan^2 \frac{\pi}{6} = 6 }$ not true?

A: The expression is not true because the sum of the values of $\cot^2 \frac{\pi}{6}$, $\operatorname{cosec} \frac{5\pi}{6}$, and $3 \tan^2 \frac{\pi}{6}$ is $\frac{1}{3} + 2 + 1 = \frac{10}{3}$, which is not equal to 6.

Common Mistakes

Here are some common mistakes to avoid when evaluating the expression:

• Not using the correct values of the trigonometric functions: Make sure to use the correct values of the trigonometric functions, such as $\cot \frac{\pi}{6} = \frac{1}{\sqrt{3}}$ and $\sin \frac{5\pi}{6} = \frac{1}{2}$.
• Not simplifying the expression correctly: Make sure to simplify the expression correctly by combining like terms and using the correct order of operations.
• Not checking the final answer: Make sure to check the final answer to ensure that it is correct.

Conclusion

Evaluating the expression ${ \cot^2 \frac{\pi}{6} + \operatorname{cosec} \frac{5\pi}{6} + 3 \tan^2 \frac{\pi}{6} = 6 }$ requires careful use of trigonometric functions and correct simplification of the expression. By avoiding common mistakes and using the correct values of the trigonometric functions, you can ensure that your final answer is correct.

Final Answer

The final answer is $\boxed{\frac{10}{3}}$.