Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.cos θ = 0.85536428

Verified step by step guidance

1. Step 1: Understand that you need to find an angle \( \theta \) such that \( \cos \theta = 0.85536428 \).
2. Step 2: Use the inverse cosine function to find \( \theta \). This is done by calculating \( \theta = \cos^{-1}(0.85536428) \).
3. Step 3: Ensure your calculator is set to degree mode since the answer needs to be in decimal degrees.
4. Step 4: Input the value 0.85536428 into your calculator and apply the inverse cosine function.
5. Step 5: The calculator will provide the angle \( \theta \) in degrees, which should be rounded to six decimal places as required.

Key Concepts

Cosine Function

The cosine function, denoted as cos(θ), is a fundamental trigonometric function that relates the angle θ in a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is defined for all angles and is periodic, with a range of values between -1 and 1. Understanding the cosine function is essential for solving problems involving angles and their corresponding ratios.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccos or cos⁻¹, are used to find the angle when the cosine value is known. For example, if cos(θ) = 0.85536428, then θ can be found using θ = cos⁻¹(0.85536428). These functions are crucial for determining angles in various applications, especially when working within specific intervals like [0°, 90°).

Angle Measurement in Degrees

Angles can be measured in degrees, with a full circle comprising 360 degrees. In this context, the interval [0°, 90°) refers to the first quadrant of the unit circle, where both sine and cosine values are positive. Understanding how to convert and interpret angles in degrees is vital for accurately solving trigonometric equations and applying them in real-world scenarios.