Evaluate The Following Trigonometric Expression Using The Principal Value For The Tangent:${ \cos \left(\tan^{-1} \sqrt{3}\right) }$A. { \frac{\sqrt{2}}{2}$}$ B. { \frac{\sqrt{3}}{3}$}$ C. 1 D. { \frac{1}{2}$}$

Introduction

Trigonometric functions are essential in mathematics, and understanding their properties is crucial for solving various problems in mathematics and physics. In this article, we will evaluate a trigonometric expression using the principal value for tangent. The expression is $\cos \left(\tan^{-1} \sqrt{3}\right)$, and we need to find its value among the given options.

Understanding the Principal Value of Tangent

The principal value of tangent is the value of the angle whose tangent is a given number. It is denoted by $\tan^{-1} x$ and is defined as the angle $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\tan \theta = x$. In this case, we have $\tan^{-1} \sqrt{3}$, which means we need to find the angle whose tangent is $\sqrt{3}$.

Finding the Angle Whose Tangent is $\sqrt{3}$

We know that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side in a right-angled triangle. Let's consider a right-angled triangle with an angle $\theta$ such that $\tan \theta = \sqrt{3}$. We can draw a diagram to represent this triangle.

  +---------------+
  |               |
  |  Opposite    |
  |  side       |
  |  (height)   |
  +---------------+
  |               |
  |  Hypotenuse  |
  |  (base)     |
  |  (length)   |
  +---------------+
  |               |
  |  Adjacent   |
  |  side       |
  |  (width)    |
  +---------------+

In this triangle, the opposite side is the height, and the adjacent side is the width. Since the tangent of the angle is $\sqrt{3}$, we can write:

$$\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\text{height}}{\text{width}} = \sqrt{3}$$

We can simplify this equation by assuming that the adjacent side is 1 unit. Then, the opposite side would be $\sqrt{3}$ units. This gives us a right-angled triangle with an angle $\theta$ such that $\tan \theta = \sqrt{3}$.

Evaluating the Trigonometric Expression

Now that we have found the angle whose tangent is $\sqrt{3}$, we can evaluate the trigonometric expression $\cos \left(\tan^{-1} \sqrt{3}\right)$. We know that the cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Let's consider the right-angled triangle we drew earlier.

  +---------------+
  |               |
  |  Opposite    |
  |  side       |
  |  (height)   |
  +---------------+
  |               |
  |  Hypotenuse  |
  |  (base)     |
  |  (length)   |
  +---------------+
  |               |
  |  Adjacent   |
  |  side       |
  |  (width)    |
  +---------------+

In this triangle, the adjacent side is 1 unit, and the hypotenuse is $\sqrt{3^2 + 1^2} = \sqrt{10}$ units. Therefore, the cosine of the angle $\theta$ is:

$$\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{1}{\sqrt{10}}$$

However, we need to find the value of $\cos \left(\tan^{-1} \sqrt{3}\right)$. Since $\tan^{-1} \sqrt{3}$ is the angle whose tangent is $\sqrt{3}$, we can write:

$$\cos \left(\tan^{-1} \sqrt{3}\right) = \cos \theta$$

where $\theta$ is the angle whose tangent is $\sqrt{3}$. Therefore, the value of $\cos \left(\tan^{-1} \sqrt{3}\right)$ is:

$$\cos \left(\tan^{-1} \sqrt{3}\right) = \frac{1}{\sqrt{10}}$$

However, this is not among the given options. Let's simplify the expression further.

Simplifying the Expression

We can simplify the expression $\cos \left(\tan^{-1} \sqrt{3}\right)$ by using the trigonometric identity:

$$\cos^2 \theta + \sin^2 \theta = 1$$

We know that $\tan \theta = \sqrt{3}$, so we can write:

$$\sin \theta = \frac{1}{\sqrt{3}}$$

Substituting this value into the trigonometric identity, we get:

$$\cos^2 \theta + \left(\frac{1}{\sqrt{3}}\right)^2 = 1$$

Simplifying this equation, we get:

$$\cos^2 \theta = \frac{2}{3}$$

Taking the square root of both sides, we get:

$$\cos \theta = \pm \sqrt{\frac{2}{3}}$$

However, we know that the cosine of an angle is always non-negative. Therefore, we can write:

$$\cos \theta = \sqrt{\frac{2}{3}}$$

Substituting this value into the expression $\cos \left(\tan^{-1} \sqrt{3}\right)$, we get:

$$\cos \left(\tan^{-1} \sqrt{3}\right) = \sqrt{\frac{2}{3}}$$

However, this is still not among the given options. Let's simplify the expression further.

Simplifying the Expression Further

We can simplify the expression $\cos \left(\tan^{-1} \sqrt{3}\right)$ by rationalizing the denominator. We can write:

$$\cos \left(\tan^{-1} \sqrt{3}\right) = \sqrt{\frac{2}{3}} = \frac{\sqrt{2}}{\sqrt{3}}$$

Rationalizing the denominator, we get:

$$\cos \left(\tan^{-1} \sqrt{3}\right) = \frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{6}}{3}$$

However, this is still not among the given options. Let's simplify the expression further.

Simplifying the Expression Further

We can simplify the expression $\cos \left(\tan^{-1} \sqrt{3}\right)$ by using the trigonometric identity:

$$\cos \left(\tan^{-1} x\right) = \frac{1}{\sqrt{1 + x^2}}$$

Substituting $x = \sqrt{3}$ into this identity, we get:

$$\cos \left(\tan^{-1} \sqrt{3}\right) = \frac{1}{\sqrt{1 + (\sqrt{3})^2}} = \frac{1}{\sqrt{4}} = \frac{1}{2}$$

Therefore, the value of $\cos \left(\tan^{-1} \sqrt{3}\right)$ is $\frac{1}{2}$.

Conclusion

In this article, we evaluated the trigonometric expression $\cos \left(\tan^{-1} \sqrt{3}\right)$ using the principal value of tangent. We found that the value of this expression is $\frac{1}{2}$. This is among the given options, and we can conclude that the correct answer is D. $\frac{1}{2}$.

Final Answer

The final answer is $\boxed{\frac{1}{2}}$.

Introduction

In our previous article, we evaluated the trigonometric expression $\cos \left(\tan^{-1} \sqrt{3}\right)$ using the principal value of tangent. We found that the value of this expression is $\frac{1}{2}$. In this article, we will answer some frequently asked questions related to evaluating trigonometric expressions using the principal value of tangent.

Q: What is the principal value of tangent?

A: The principal value of tangent is the value of the angle whose tangent is a given number. It is denoted by $\tan^{-1} x$ and is defined as the angle $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\tan \theta = x$.

Q: How do I find the principal value of tangent?

A: To find the principal value of tangent, you can use a calculator or a trigonometric table. Alternatively, you can use the following formula:

$$\tan^{-1} x = \frac{1}{2} \ln \left(\frac{1 + x}{1 - x}\right)$$

Q: What is the range of the principal value of tangent?

A: The range of the principal value of tangent is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Q: How do I evaluate a trigonometric expression using the principal value of tangent?

A: To evaluate a trigonometric expression using the principal value of tangent, you can use the following steps:

1. Find the principal value of tangent using the formula $\tan^{-1} x$.
2. Substitute the principal value of tangent into the trigonometric expression.
3. Simplify the expression using trigonometric identities.

Q: What are some common trigonometric identities that I can use to simplify expressions?

A: Some common trigonometric identities that you can use to simplify expressions include:

• $\sin^2 \theta + \cos^2 \theta = 1$
• $\cos^2 \theta + \sin^2 \theta = 1$
• $\tan^2 \theta + 1 = \sec^2 \theta$
• $\cot^2 \theta + 1 = \csc^2 \theta$

Q: How do I rationalize the denominator of a fraction?

A: To rationalize the denominator of a fraction, you can multiply the numerator and denominator by the conjugate of the denominator. For example, if you have a fraction $\frac{a}{b + c}$, you can rationalize the denominator by multiplying the numerator and denominator by $b - c$.

Q: What is the final answer to the trigonometric expression $\cos \left(\tan^{-1} \sqrt{3}\right)$?

A: The final answer to the trigonometric expression $\cos \left(\tan^{-1} \sqrt{3}\right)$ is $\frac{1}{2}$.

Conclusion

In this article, we answered some frequently asked questions related to evaluating trigonometric expressions using the principal value of tangent. We hope that this article has been helpful in clarifying some of the concepts and providing additional information on how to evaluate trigonometric expressions.

Final Answer

The final answer is $\boxed{\frac{1}{2}}$.