If Tan ⁡ ( Θ ) = 7 24 \tan (\theta)=\frac{7}{24} Tan ( Θ ) = 24 7 , What Is The Measure Of Θ \theta Θ ?A. 16.26 ∘ 16.26^{\circ} 16.2 6 ∘ B. 14.05 ∘ 14.05^{\circ} 14.0 5 ∘ C. 22.2 ∘ 22.2^{\circ} 22. 2 ∘ D. 18.74 ∘ 18.74^{\circ} 18.7 4 ∘

Mar 13, 2025 by ADMIN 238 views

**Solving for the Measure of an Angle in a Right Triangle**

Introduction

In trigonometry, the tangent function is used to relate the angles and side lengths of a right triangle. Given the tangent of an angle, we can use this relationship to find the measure of the angle itself. In this article, we will explore how to solve for the measure of an angle in a right triangle when the tangent of the angle is known.

Understanding the Tangent Function

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

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

In the given problem, we are told that $\tan (\theta) = \frac{7}{24}$ . This means that the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle is $\frac{7}{24}$ .

Using the Arctangent Function to Solve for the Angle

To find the measure of the angle, we can use the arctangent function, which is the inverse of the tangent function. The arctangent function returns the angle whose tangent is a given number. In this case, we want to find the angle whose tangent is $\frac{7}{24}$ .

Using a calculator or a trigonometric table, we can find that the arctangent of $\frac{7}{24}$ is approximately $16.26^{\circ}$ .

Verifying the Answer

To verify that this is the correct answer, we can use the tangent function to check if the ratio of the side opposite the angle to the side adjacent to the angle is indeed $\frac{7}{24}$ .

Let's assume that the side opposite the angle is 7 units and the side adjacent to the angle is 24 units. Using the tangent function, we can calculate the ratio of the side opposite the angle to the side adjacent to the angle:

\tan (\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{7}{24}

This confirms that the ratio of the side opposite the angle to the side adjacent to the angle is indeed $\frac{7}{24}$ , and therefore the measure of the angle is approximately $16.26^{\circ}$ .

Conclusion

In this article, we have shown how to solve for the measure of an angle in a right triangle when the tangent of the angle is known. By using the arctangent function, we can find the angle whose tangent is a given number. We have also verified that the answer is correct by using the tangent function to check if the ratio of the side opposite the angle to the side adjacent to the angle is indeed the given number.

Answer

The final answer is $\boxed{16.26^{\circ}}$ .

Additional Information

The tangent function is used to relate the angles and side lengths of a right triangle.
The arctangent function is the inverse of the tangent function and returns the angle whose tangent is a given number.
To verify that the answer is correct, we can use the tangent function to check if the ratio of the side opposite the angle to the side adjacent to the angle is indeed the given number.

References

Q: What is the tangent function and how is it used in trigonometry?

A: The tangent function is a trigonometric function that relates the angles and side lengths of a right triangle. It is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The tangent function is used to solve for the measure of an angle in a right triangle when the ratio of the side opposite the angle to the side adjacent to the angle is known.

Q: How do I use the arctangent function to solve for the measure of an angle in a right triangle?

A: To use the arctangent function to solve for the measure of an angle in a right triangle, you need to know the ratio of the side opposite the angle to the side adjacent to the angle. You can then use a calculator or a trigonometric table to find the angle whose tangent is the given ratio. The arctangent function returns the angle in degrees or radians.

Q: What is the difference between the tangent function and the arctangent function?

A: The tangent function is used to relate the angles and side lengths of a right triangle, while the arctangent function is the inverse of the tangent function and returns the angle whose tangent is a given number.

Q: How do I verify that the answer I get from using the arctangent function is correct?

A: To verify that the answer you get from using the arctangent function is correct, you can use the tangent function to check if the ratio of the side opposite the angle to the side adjacent to the angle is indeed the given ratio. This will confirm that the measure of the angle is correct.

Q: Can I use the tangent function to solve for the measure of an angle in a right triangle if I know the lengths of the sides of the triangle?

A: Yes, you can use the tangent function to solve for the measure of an angle in a right triangle if you know the lengths of the sides of the triangle. You can use the tangent function to find the ratio of the side opposite the angle to the side adjacent to the angle, and then use the arctangent function to find the angle.

Q: What are some common mistakes to avoid when using the tangent function and the arctangent function?

A: Some common mistakes to avoid when using the tangent function and the arctangent function include:

Using the wrong ratio of the side opposite the angle to the side adjacent to the angle.
Using the wrong unit of measurement (e.g. degrees vs. radians).
Not verifying the answer using the tangent function.
Not using the arctangent function to find the angle.

Q: Can I use the tangent function and the arctangent function to solve for the measure of an angle in a right triangle if the triangle is not a right triangle?

A: No, the tangent function and the arctangent function are only used to solve for the measure of an angle in a right triangle. If the triangle is not a right triangle, you will need to use other trigonometric functions and techniques to solve for the measure of the angle.

Q: Are there any other trigonometric functions that I can use to solve for the measure of an angle in a right triangle?

A: Yes, there are other trigonometric functions that you can use to solve for the measure of an angle in a right triangle, including the sine function and the cosine function. However, the tangent function and the arctangent function are the most commonly used functions for this purpose.

Q: Can I use a calculator to solve for the measure of an angle in a right triangle?

A: Yes, you can use a calculator to solve for the measure of an angle in a right triangle. Most calculators have a tangent function and an arctangent function that you can use to solve for the measure of an angle.

Q: Are there any online resources that I can use to learn more about solving for the measure of an angle in a right triangle?

A: Yes, there are many online resources that you can use to learn more about solving for the measure of an angle in a right triangle, including online tutorials, videos, and practice problems. Some popular online resources include Khan Academy, Mathway, and Wolfram Alpha.