Use a calculator to approximate each value in decimal degrees.
θ = cos⁻¹ 0.80396577

Verified step by step guidance

1. Understand that \( \theta = \cos^{-1}(x) \) is the inverse cosine function, which gives the angle \( \theta \) whose cosine is \( x \).
2. Identify the given value: \( x = 0.80396577 \).
3. Use a calculator to find \( \theta = \cos^{-1}(0.80396577) \).
4. Ensure the calculator is set to degree mode to get the angle in decimal degrees.
5. Read the calculator output to find the approximate value of \( \theta \) in decimal degrees.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹ (arccosine), are used to find the angle whose cosine is a given value. For example, if cos(θ) = x, then θ = cos⁻¹(x). These functions are essential for solving problems where the angle is unknown and must be determined from a known ratio.

Calculator Functions

Using a scientific or graphing calculator effectively is crucial for approximating trigonometric values. Most calculators have specific modes for degrees and radians, and it's important to ensure the calculator is set to the correct mode when calculating angles. This ensures that the output is in the desired unit of measurement.

Understanding Cosine Values

The cosine function relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. The value of cos(θ) ranges from -1 to 1, and understanding this range helps in interpreting the results of inverse cosine calculations. Knowing that a cosine value of 0.80396577 corresponds to a specific angle aids in visualizing the problem geometrically.