Use a calculator to approximate each value in decimal degrees.
θ = arccos (-0.39876459)

Verified step by step guidance

1. Understand that the problem asks for the angle \( \theta \) whose cosine value is \( -0.39876459 \). This is expressed as \( \theta = \arccos(-0.39876459) \).
2. Recall that the \( \arccos \) function (inverse cosine) returns an angle in the range \( 0^\circ \) to \( 180^\circ \) when working in degrees.
3. Use a calculator set to degree mode to find the approximate value of \( \theta \). Input the value \( -0.39876459 \) into the \( \arccos \) function.
4. The calculator will provide the angle \( \theta \) in decimal degrees, which is the angle whose cosine is \( -0.39876459 \).
5. Interpret the result as the principal value of the angle in degrees, which completes the approximation.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccos, are used to find the angle that corresponds to a given trigonometric ratio. For arccos(x), it returns the angle whose cosine is x, typically within the range 0° to 180° for degrees.

Domain and Range of arccos

The arccos function accepts input values between -1 and 1, inclusive, since cosine values lie within this range. Its output, or range, is limited to angles between 0° and 180°, ensuring a unique principal value for each input.

Using a Calculator for Trigonometric Approximations

Calculators can compute inverse trig functions to approximate angle measures in decimal degrees. It is important to set the calculator to degree mode and input the value correctly to obtain an accurate angle approximation.