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

Verified step by step guidance

1. Identify the problem: You need to find the angle \( \theta \) such that \( \cos(\theta) = 0.80396577 \). This means \( \theta = \cos^{-1}(0.80396577) \).
2. Recall that the inverse cosine function, \( \cos^{-1} \), returns an angle in radians or degrees depending on your calculator settings. Since the problem asks for decimal degrees, ensure your calculator is set to degree mode.
3. Input the value \( 0.80396577 \) into your calculator and use the inverse cosine function (often labeled as \( \cos^{-1} \) or \( \arccos \)) to find \( \theta \).
4. The calculator will provide an approximate decimal value for \( \theta \) in degrees. This is the angle whose cosine is \( 0.80396577 \).
5. Interpret the result: \( \theta \) is the principal value of the angle in degrees between 0° and 180° because the range of \( \cos^{-1} \) is \( [0^\circ, 180^\circ] \).

Key Concepts

Inverse Cosine Function (cos⁻¹)

The inverse cosine function, denoted as cos⁻¹ or arccos, returns the angle whose cosine value is a given number. It is used to find an angle when the cosine value is known, with the output typically in the range of 0° to 180° for real numbers.

Using a Calculator for Trigonometric Functions

Calculators can compute inverse trigonometric functions to provide angle measures in degrees or radians. It is important to ensure the calculator is set to the correct mode (degrees in this case) before calculating the inverse cosine to get the desired unit.

Decimal Degree Approximation

Decimal degrees express angles as a decimal number rather than degrees, minutes, and seconds. This format is useful for precise calculations and is commonly used in scientific and engineering contexts for clarity and ease of computation.