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: \( \cos^{-1} \left( \frac{\sqrt{5}}{7} \right) \). This represents the inverse cosine function, which will give us an angle whose cosine is \( \frac{\sqrt{5}}{7} \).
2. Ensure that the value \( \frac{\sqrt{5}}{7} \) is within the valid range for the inverse cosine function, which is \([-1, 1]\).
3. Use a calculator to compute \( \cos^{-1} \left( \frac{\sqrt{5}}{7} \right) \). Make sure your calculator is set to the correct mode (degrees or radians) as required by the context of your problem.
4. Round the result from the calculator to two decimal places.
5. Verify the result by checking if the cosine of the calculated angle returns approximately \( \frac{\sqrt{5}}{7} \).

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. These functions are essential for solving problems where the angle is unknown, and they return values in a specific range, typically between 0 and π for cosine.

Domain and Range of Cosine

The cosine function has a domain of all real numbers and a range of [-1, 1]. For the inverse cosine function, the input must be within this range. Understanding the domain and range is crucial for determining valid inputs when using inverse trigonometric functions.

Calculator Functions

Using a calculator to evaluate trigonometric functions requires familiarity with its settings, particularly whether it is in degree or radian mode. For the expression cos⁻¹(√5/7), it is important to ensure the calculator is set to the correct mode to obtain the angle in the desired unit, rounded to two decimal places.