What Is $\tan \left(16^{\circ}\right$\]?A. 0.29 B. 0.96 C. 0.16 D. 0.39

$$

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the fundamental trigonometric functions is the tangent function, denoted by $\tan$. In this article, we will explore the value of \tan \left(16^{\circ}\right].

Understanding the Tangent Function

The tangent function is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle in a right-angled triangle. Mathematically, it can be expressed as:

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$

where $\theta$ is the angle being considered.

Calculating \tan \left(16^{\circ}\right]

To calculate the value of \tan \left(16^{\circ}\right], we can use a calculator or a trigonometric table. However, in this article, we will use a more mathematical approach to derive the value.

One way to calculate the value of \tan \left(16^{\circ}\right] is to use the half-angle formula for tangent:

$$\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)}$$

We can rewrite this formula as:

$$\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)} = \frac{1 - \cos(32^{\circ})}{\sin(32^{\circ})}$$

Using a calculator or a trigonometric table, we can find the values of $\cos(32^{\circ})$ and $\sin(32^{\circ})$:

$$\cos(32^{\circ}) \approx 0.8486$$

$$\sin(32^{\circ}) \approx 0.5299$$

Substituting these values into the formula, we get:

$$\tan\left(\frac{16^{\circ}}{2}\right) = \frac{1 - 0.8486}{0.5299} \approx 0.196$$

However, this is not the value of \tan \left(16^{\circ}\right]. We need to use the double-angle formula for tangent to get the correct value:

$$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$$

We can rewrite this formula as:

$$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} = \frac{2\tan(8^{\circ})}{1 - \tan^2(8^{\circ})}$$

Using a calculator or a trigonometric table, we can find the value of $\tan(8^{\circ})$:

$$\tan(8^{\circ}) \approx 0.1406$$

Substituting this value into the formula, we get:

$$\tan(16^{\circ}) = \frac{2\tan(8^{\circ})}{1 - \tan^2(8^{\circ})} \approx \frac{2(0.1406)}{1 - (0.1406)^2} \approx 0.296$$

Conclusion

In this article, we have explored the value of \tan \left(16^{\circ}\right]. We have used the half-angle formula and the double-angle formula for tangent to derive the value. The final answer is approximately 0.296.

Answer

The correct answer is A. 0.29.

References

• "Trigonometry" by Michael Corral
• "Calculus" by Michael Spivak
• "Trigonometry for Dummies" by Mary Jane Sterling

Note

$$ $$

Introduction

In our previous article, we explored the value of \tan \left(16^{\circ}\right]. In this article, we will answer some frequently asked questions related to the tangent function and its applications.

Q: What is the tangent function?

A: The tangent function is a fundamental trigonometric function that deals with the relationships between the sides and angles of triangles. It is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle in a right-angled triangle.

Q: How do I calculate the value of \tan \left(16^{\circ}\right]?

A: You can use a calculator or a trigonometric table to calculate the value of \tan \left(16^{\circ}\right]. Alternatively, you can use the half-angle formula and the double-angle formula for tangent to derive the value.

Q: What is the half-angle formula for tangent?

A: The half-angle formula for tangent is:

$$\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)}$$

Q: What is the double-angle formula for tangent?

A: The double-angle formula for tangent is:

$$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$$

Q: How do I use the half-angle formula and the double-angle formula for tangent?

A: To use the half-angle formula and the double-angle formula for tangent, you need to substitute the values of $\theta$ and the trigonometric functions into the formulas. For example, to calculate the value of \tan \left(16^{\circ}\right], you can use the half-angle formula and the double-angle formula for tangent.

Q: What are some real-world applications of the tangent function?

A: The tangent function has many real-world applications, including:

• Navigation: The tangent function is used in navigation to calculate the direction and distance between two points.
• Physics: The tangent function is used in physics to describe the motion of objects and the forces acting on them.
• Engineering: The tangent function is used in engineering to design and analyze structures, such as bridges and buildings.

Q: How do I use a calculator to calculate the value of \tan \left(16^{\circ}\right]?

A: To use a calculator to calculate the value of \tan \left(16^{\circ}\right], you need to enter the value of $\theta$ and the calculator will display the value of \tan \left(\theta\right].

Q: What are some common mistakes to avoid when calculating the value of \tan \left(16^{\circ}\right]?

A: Some common mistakes to avoid when calculating the value of \tan \left(16^{\circ}\right] include:

• Using the wrong formula or formulae.
• Making errors in the calculation.
• Not checking the units of the answer.

Conclusion

In this article, we have answered some frequently asked questions related to the tangent function and its applications. We hope that this article has been helpful in clarifying any doubts you may have had about the tangent function.

References

• "Trigonometry" by Michael Corral
• "Calculus" by Michael Spivak
• "Trigonometry for Dummies" by Mary Jane Sterling

Note

The tangent function is a fundamental concept in mathematics and has many real-world applications. It is essential to understand the tangent function and its applications to solve problems in mathematics and other fields.