Use a calculator to approximate each value in decimal degrees.
θ = arctan 1.7804675

Verified step by step guidance

1. Understand that \( \theta = \arctan(1.7804675) \) means you are looking for an angle \( \theta \) whose tangent is 1.7804675.
2. Recall that the arctangent function, \( \arctan(x) \), is the inverse of the tangent function, \( \tan(\theta) \). It returns the angle whose tangent is \( x \).
3. Use a calculator to find \( \theta \) by inputting \( \arctan(1.7804675) \). Ensure your calculator is set to degree mode to get the angle in decimal degrees.
4. The calculator will provide an angle \( \theta \) in degrees, which is the solution to the problem.
5. Verify the result by checking if \( \tan(\theta) \approx 1.7804675 \) using the calculated angle to ensure accuracy.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, are used to find the angle whose tangent is a given number. In this case, arctan(1.7804675) will yield the angle θ in radians or degrees, depending on the calculator settings. Understanding how to use these functions is essential for solving problems involving angles and their corresponding trigonometric ratios.

Tangent Function

The tangent function is one of the primary trigonometric functions, defined as the ratio of the opposite side to the adjacent side in a right triangle. It is crucial to understand that the tangent of an angle can take any real number value, which is why the arctan function can return angles for both positive and negative inputs. This relationship is foundational in trigonometry and helps in visualizing angle measures.

Calculator Settings

When using a calculator for trigonometric functions, it is important to ensure that the calculator is set to the correct mode: degrees or radians. The output of functions like arctan will vary based on this setting, affecting the interpretation of the result. For this question, since the output is requested in decimal degrees, the calculator must be set to degree mode to obtain the correct angle measurement.