In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places.cos⁻¹ 3/8

Verified step by step guidance

1. Understand that \( \cos^{-1} \) is the inverse cosine function, which gives the angle whose cosine is the given value.
2. Set up the equation \( \theta = \cos^{-1} \left( \frac{3}{8} \right) \), where \( \theta \) is the angle we need to find.
3. Use a calculator to compute \( \cos^{-1} \left( \frac{3}{8} \right) \). Ensure the calculator is in the correct mode (degrees or radians) as required by the context of the problem.
4. Round the result from the calculator to two decimal places.
5. Verify the result by checking if \( \cos(\theta) \approx \frac{3}{8} \) using the rounded value of \( \theta \).

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. For example, if cos(θ) = x, then θ = cos⁻¹(x). These functions are essential for solving problems where the angle is unknown and can be applied in various contexts, including geometry and physics.

Domain and Range of Cosine

The cosine function has a domain of all real numbers and a range of [-1, 1]. This means that for any valid input to the cosine function, the output will always fall within this range. Consequently, when using the inverse cosine function, the input must also be within [-1, 1] to yield a real angle, which is crucial for determining valid expressions.

Calculator Functions and Rounding

Using a calculator to evaluate trigonometric functions requires understanding how to input values correctly and interpret the results. After calculating the inverse cosine, rounding to two decimal places is often necessary for clarity and precision in reporting results. Familiarity with calculator functions ensures accurate computations and proper formatting of answers.