In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. _ cos⁻¹ √5/7

Verified step by step guidance

1. Identify the expression to evaluate: \(\cos^{-1} \left( \frac{\sqrt{5}}{7} \right)\), which means finding the angle whose cosine is \(\frac{\sqrt{5}}{7}\).
2. Calculate the value inside the inverse cosine function: first find \(\sqrt{5}\), then divide it by 7 to get the numerical value of the cosine.
3. Use a calculator set to degree mode (or radian mode, depending on the problem's requirement) to find the angle \(\theta = \cos^{-1} \left( \frac{\sqrt{5}}{7} \right)\).
4. Round the resulting angle to two decimal places as requested.
5. Interpret the result as the angle in degrees (or radians) whose cosine is \(\frac{\sqrt{5}}{7}\), completing the evaluation.

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 of that angle is known, with the output typically in radians or degrees within the range 0 to π (0° to 180°).

Evaluating Square Roots and Fractions

Understanding how to simplify and evaluate expressions involving square roots and fractions is essential. Here, √5/7 means the square root of 5 divided by 7, which must be calculated accurately before applying the inverse cosine function.

Using a Calculator for Trigonometric Functions

Calculators can compute inverse trigonometric functions and handle decimal approximations. Knowing how to input values correctly and round the result to a specified number of decimal places, such as two decimals, is crucial for obtaining the final answer.