Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 is 0.624.

Step 1: Find the inverse tangent

First, calculate the value of \( \tan^{-1}(2.001) \). Using a calculator, input 2.001 and find its arctangent or inverse tangent. This will give you an angle \( \theta \) in radians. Let's denote this angle as \( \theta = \tan^{-1}(2.001) \approx 1.107 \) radians.

Step 2: Calculate the sine of the angle

Next, determine the sine of the angle that we found in the previous step. Compute \( \sin(\theta) \), where \( \theta \approx 1.107 \). Again, use a calculator to find the sine value. This gives us \( \sin(1.107) \approx 0.894 \).

Step 3: Calculate the cosine of the sine value

Finally, calculate the cosine of the sine value obtained in step 2. Input \( 0.894 \) into your calculator to find \( \cos(0.894) \). This calculation will provide the final result of the expression. The approximation is \( \cos(0.894) \approx 0.624 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are an important aspect of trigonometry. They allow us to find an angle given a trigonometric ratio. For example, the inverse tangent function, denoted as \( \tan^{-1} \), gives us the angle whose tangent value is a given number.
Some key points to remember about inverse functions include:

• The output of inverse trigonometric functions is an angle, usually in radians or degrees.
• These functions "undo" the regular trigonometric functions.
• Calculators typically provide outputs in radians by default.

Understanding inverse trigonometric functions helps solve problems involving angle measurements based on given trigonometric values.

Angle Approximation

Angle approximation involves estimating the value of an angle based on trigonometric expressions. Accurate approximations are essential, especially when using calculators.
In the exercise, we use the inverse tangent function, \( \tan^{-1}(2.001) \), to approximate an angle, \( \theta \). The result, \( \theta \approx 1.107 \) radians, is an approximation of this value.
Keep these tips in mind:

• Rounding can lead to small differences in final results.
• Being precise with calculator settings mitigates errors.
• Understanding radians and degrees eliminates conversion mistakes.

Angle approximation is all about finding accurate, useful approximations that can solve complex trigonometric problems.

Calculator Use

Calculators are indispensable tools in trigonometry, providing quick and accurate calculations of complex functions. Using a calculator effectively involves understanding its functions and settings.
Some essential tips for calculator use include:

• Always check if your calculator is in the correct mode (radians or degrees) for your calculation.
• Use parentheses to ensure correct order of operations, especially in nested functions.
• Be familiar with calculator-specific trigonometric function buttons for accurate input.

Efficient calculator use helps ensure accurate and reliable results in trigonometric computations.

Cosine

Cosine is one of the primary trigonometric functions. It measures the ratio of the adjacent side to the hypotenuse in a right-angled triangle, but is also defined for all real numbers using the unit circle.
In our exercise, after finding \( \sin(\theta) \approx 0.894 \), we calculate \( \cos(0.894) \) using a calculator. The cosine function helps determine this value, where \( \cos(0.894) \approx 0.624 \).
Key characteristics of cosine include:

• Cosine values range between -1 and 1.
• It is periodic with a period of \(2\pi\) radians.
• Cosine is even, meaning \(\cos(-x) = \cos(x)\).

Understanding cosine is crucial for solving diverse trigonometric problems efficiently.

Sine

The sine function is another fundamental trigonometric function. It represents the ratio of the opposite side to the hypotenuse in a right-angled triangle, and is similarly defined over the unit circle for all angles.
For this exercise, calculating \( \sin(\theta) \) where \( \theta \approx 1.107 \) provided us with \( \sin(1.107) \approx 0.894 \). This is an important step before evaluating the cosine of the result.
Some essential properties of sine include:

• Sine values also range between -1 and 1.
• It is periodic like cosine, but with a phase shift, repeating every \(2\pi\).
• Sine is odd, so \(\sin(-x) = -\sin(x)\).

A deep understanding of the sine function allows for accurate and effective trigonometric calculations.