Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 11-18, use a calculator to approximate each value. \(\cos \left(\sin \left(\tan ^{-1} 2.001\right)\right)\)

Step-by-Step Solution

Answer

The approximate value of the expression is 0.6270.

Step 1: Calculate the Inverse Tangent

First, find the inverse tangent (arctan) of 2.001 using a calculator. This step will help determine the angle whose tangent is 2.001. Use the calculator function: \[tan^{-1}(2.001)\approx 63.4349^\circ\] Convert this angle to radians because most calculators use radians for trigonometric functions: \[63.4349^\circ \approx 1.1071\, \text{radians}\]

Step 2: Take the Sine of the Result

Next, calculate the sine of the angle obtained in Step 1. Continue using the radians value for accuracy.Using the calculator, compute:\[\sin(1.1071) \approx 0.8945\]

Step 3: Calculate the Cosine of the Sine Result

Lastly, find the cosine of the result obtained in Step 2.Use the calculator to evaluate:\[\cos(0.8945) \approx 0.6270\]

Step 4: Final Result

So, using the calculator and following the steps shown, the approximate value of the expression is:\[\cos \left(\sin \left(\tan ^{-1} 2.001\right)\right) \approx 0.6270\]

Key Concepts

Inverse Trigonometric Functions

When we discuss inverse trigonometric functions, we are diving into the world of finding angles based on the values of trigonometric functions. In the given problem, the function \( an^{-1}\) is used, also known as the arctangent function. This function helps us determine an angle whose tangent value is a specific number.
For example, \( an^{-1}(2.001)\) asks: "What angle has a tangent of 2.001?". Calculating this, we find it approximately equals \( an^{-1}(2.001) \approx 63.4349^\circ\). It's important to note that inverse trigonometric functions often return values within a restricted range to maintain uniqueness.

• Arctangent values are typically provided in the range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians.
• This ensures that each output value corresponds to a unique input angle in trigonometric functions.

Angle Conversion

In trigonometry calculations, it's crucial to work with angles correctly, and sometimes this means converting angles from degrees to radians or vice-versa. Many calculators are set up to use radians by default for trigonometric calculations.
To convert from degrees to radians, the formula is:

• \[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]

Applying this to our problem, the calculated angle of \(63.4349^\circ\) is converted to radians to maintain accuracy in subsequent steps:

• \(63.4349^\circ \approx 1.1071 \, \text{radians}\)

Understanding how to convert between these two units of angle measurement is fundamental when handling problems involving trigonometric functions.

Calculation Steps

To solve the given expression, let's break down each calculation step by step. This approach simplifies complex problems and prevents errors.

1. **Find the Inverse Tangent:** - Use a calculator to compute \(\tan^{-1}(2.001) = 63.4349^\circ\). - Convert this angle to radians: \(\approx 1.1071\, \text{radians}\).
2. **Compute the Sine:** - With the angle in radians, compute the sine of the angle: \(\sin(1.1071) \approx 0.8945\).
3. **Calculate the Cosine:** - Finally, take the cosine of the sine result: \(\cos(0.8945) \approx 0.6270\).
These steps show the logical progression from finding an angle using inverse functions, converting it, and then applying further trigonometric functions to reach the final answer.

Calculator Usage

Using a calculator effectively can save time and increase accuracy for trigonometry problems. Most scientific calculators today offer functions for direct trigonometric and inverse trigonometric calculations.
Here’s how you can use your calculator to solve these types of problems:

• Ensure your calculator is set to the correct angle unit before starting. Use radians for this problem.
• Use the \(\tan^{-1}\) function directly to find the inverse tangent, entering the value within brackets.
• After calculating the inverse function, switch to the sine \(\sin\) function, again using the radians result from the previous step.
• Finally, employ the cosine \(\cos\) function on the obtained sine value.

By methodically setting your calculator with these steps, you can ensure precision and confidence in your answers. Always double-check if you're unsure which angle mode to use by referring to the problem's instructions or context.