Problem 13 - Algebra and Trigonometry

Question

\(11-22\) . Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. \(\cos ^{-1}\left(-\frac{3}{7}\right)\)

Step-by-Step Solution

Answer

The approximate value is 1.99315 radians.

Step 1: Understand the Problem

We are asked to find the approximate value of the inverse cosine (arccos) of \(-\frac{3}{7}\), correct to five decimal places. This expression is defined because the input is within the range of cosine function, which is between -1 and 1.

Step 2: Setup the Expression for Calculation

We need to compute \(\cos^{-1}\left(-\frac{3}{7}\right)\). This is the angle whose cosine is \(-\frac{3}{7}\). To calculate this, we will use a scientific calculator or software that can compute inverse trigonometric functions.

Step 3: Calculate the Inverse Cosine

Using a calculator, enter the value \(-\frac{3}{7}\) into the inverse cosine function. Ensure the calculator is set to give the result in radians, as this is the standard unless specified otherwise.

Step 4: Rounding the Result

After obtaining the result from the calculator, round it to five decimal places. Keep all rounding rules in mind, ensuring that digits beyond the fifth decimal place are not used to alter the value at the fifth position.

Key Concepts

The Arccos Function

The arccos function, often denoted as \( \cos^{-1}(x) \), is the inverse of the cosine function. This function gives us the angle whose cosine is the specified number. When we're finding \( \cos^{-1}(x) \), what we're really doing is asking: "What angle, when you take the cosine of it, gives us this specific value?" This function is a crucial concept in trigonometry, especially for solving problems involving angles and triangles.

One important point about the arccos function is its range. The output of \( \cos^{-1}(x) \) is always an angle. In mathematical terms, this is within the range of 0 to \( \pi \) radians. This means that any value you get from the arccos function will be an angle between 0 and 180 degrees. This is different from other trigonometric functions like sine, which have outputs in different ranges.

Using the arccos function on a calculator simplifies finding angles from cosine values that might seem difficult to solve otherwise. Make sure the calculator is in radian mode unless stated to be in degrees.

Understanding the Cosine Range

The range of the cosine function is very unique and important to remember. It ranges from -1 to 1. This means any cosine value must be within this interval. Values outside this range have no corresponding angle in real-world trigonometry and thus are considered undefined.

For instance, if we're given a cosine value of -3/7, like in our exercise, it is perfectly valid because -3/7 is within the -1 to 1 range. This ensures that when we use the \( \cos^{-1}(-\frac{3}{7}) \), we indeed can find a real angle corresponding to this cosine value.

Always check whether cosine values lie within the permissible range before performing calculations. This will prevent errors and ensure the calculations have a meaningful outcome.

The Importance of Rounding Decimals

Rounding decimals is vital in mathematics to give a clean and precise representation of numbers, especially when dealing with trigonometric functions and their outcomes. When the result of a calculation includes a long decimal, rounding it makes it more manageable and easier to interpret.

In our step by step solution, we were required to round the inverse cosine result to five decimal places. Rounding to five decimal places ensures consistency and accuracy in problems where precision is needed. Here’s a quick reminder about rounding rules:

• If the digit right after your desired precision is 5 or more, round the last kept digit up by 1.
• If it’s less than 5, keep the last kept digit the same.

By following these rules, you ensure your value represents as much accuracy as possible within the required precision. This practice is often crucial in physics and engineering problems, where precise calculations can impact the entire outcome of a solution.