Problem 9 - Algebra and Trigonometry

Question

Use a calculator to find an approximate value of each expression rounded to five decimal places, if it is defined. $$ \cos ^{-1}\left(-\frac{1}{4}\right) $$

Step-by-Step Solution

Answer

\( \cos^{-1}(-\frac{1}{4}) \approx 1.82348 \) radians.

Step 1: Understand the Inverse Cosine Function

The inverse cosine function, written as \( \cos^{-1}(x) \), returns the angle whose cosine is \( x \). The range of the inverse cosine function is \([0, \pi]\) radians.

Step 2: Identify the Input Value

In this exercise, the input value to the inverse cosine function is \(-\frac{1}{4}\). This value indicates we need to find the angle where the cosine is \(-\frac{1}{4}\).

Step 3: Input the Value Into the Calculator

Use a scientific calculator to compute \( \cos^{-1}(-\frac{1}{4}) \). Ensure that your calculator is set to the correct mode (radians, in this case) to get an accurate result.

Step 4: Round the Result

After calculating \( \cos^{-1}(-\frac{1}{4}) \), round the result to five decimal places. The calculated value is approximately 1.82348 radians.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental tools in mathematics, particularly useful for studying triangles and modeling periodic phenomena. Among them, cosine, sine, and tangent are the most commonly used. The cosine function (\( \cos \theta \)) relates the angle \( \theta \) of a right triangle to the ratio of the length of the adjacent side to the hypotenuse.
Inverse trigonometric functions, like the inverse cosine or \( \cos^{-1}(x) \), work in reverse. They allow us to find the angle when given a trigonometric ratio. In our example, \( \cos^{-1}(-\frac{1}{4}) \) finds an angle where the cosine is \(-\frac{1}{4}\). This function has a range of \([0, \pi]\)
Knowing the inverse function's range is essential, as it ensures the angle is within a specific interval, making the result unique and predictable. So whenever you encounter inverse trigonometric functions, remember they help you move from a ratio back to an angle.

Calculating Angles

Calculating angles using inverse trigonometric functions involves a few straightforward steps. Here's a simple guide to follow:

• First, identify the trigonometric function and value. For instance, if you have \( \cos^{-1}(x) \), you know you're dealing with cosine.
• Next, consider the function's range. The inverse cosine function's range, as discussed, is \([0, \pi]\)
• Then, use a calculator to input the value and determine the angle. For \( \cos^{-1}(-\frac{1}{4}) \), we're interested in finding the angle whose cosine is \(-\frac{1}{4}\)

Remember, angles in trigonometry can be represented in degrees or radians. For this exercise, we use radians, which are handy for many scientific calculations. Calculating angles requires careful attention to detail to ensure precise results, particularly when rounding.

Use of Calculators

When dealing with trigonometric functions and calculating angles, calculators become invaluable allies. They save time and reduce the complexity of tedious computations.
To find \( \cos^{-1}(-\frac{1}{4}) \) using a calculator:

• Ensure your calculator is in the correct mode: radians or degrees. Our context requires radians.
• Input the value into the calculator. Most scientific calculators have a button for inverse trigonometric functions like \( \cos^{-1} \).
• Press the appropriate sequence to perform the calculation, producing a precise result.
• Round to the specified decimal places (five, in this case) for clarity and precision.

Calculators make finding approximate values of trigonometric expressions user-friendly and accessible, even for those new to the subject. Utilizing these tools effectively ensures you have accurate and efficient results.