Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 11-18, use a calculator to approximate each value. \(\sin ^{2}(\ln (\cos 0.5555))\)

Step-by-Step Solution

Answer

The approximate value of the expression is 0.4679.

Step 1: Evaluate the cosine

Calculate \(\cos(0.5555)\) using a calculator. The cosine function measures how much of a unit circle's radius is 'covered' in the horizontal direction when rotating by that angle.

Step 2: Compute the natural logarithm

Take the result from Step 1 and calculate the natural logarithm \(\ln(x)\). This operation helps compress numbers between 0 and 1, and expand numbers greater than 1.

Step 3: Calculate the sine value

Take the result from Step 2 and compute \(\sin(x)\) using a calculator. This assesses the vertical component of an angle corresponding to the logarithm result on a unit circle.

Step 4: Square the sine result

Square the result obtained from Step 3. This will give \(\sin^{2}(x)\), representing the squared vertical component of the original expression.

Step 5: Approximate the value

Ensure all calculations are correct by re-evaluating steps as necessary and summarize the overall expression \(\sin ^{2}(\ln (\cos 0.5555))\) to find an approximate value.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, often abbreviated as "sin." It is used to determine the vertical component of a point on the unit circle corresponding to a given angle. For an angle

• The sine value ranges from -1 to 1.
• A positive sine value indicates a position above the horizontal axis.
• A negative sine value signals a position below the horizontal axis.

When computing \( \sin(x) \) where \(x\) is derived from other calculations, such as a logarithm, it represents the openness of a vertical angle. This is important in a variety of fields, such as physics and engineering, where the vertical position is crucial for calculations. Understanding the sine function's nature helps in realizing how it impacts further mathematical operations, like squaring in the given problem.

Cosine Function

The cosine function, denoted as "cos," gives the horizontal component of an angle on the unit circle. In the expression

• The angle is represented as a measure in radians.
• Cosine values also range between -1 and 1.
• A positive cosine value indicates a position to the right of the vertical axis.
• A negative cosine value indicates a position to the left.

Using a calculator to approximate \( \cos(0.5555) \) is the first step in the exercise. This value provides the foundation needed for the next steps, impacting subsequent results significantly. The cosine's link to angles and the unit circle makes it an indispensable tool in trigonometry for modeling periodic phenomena.

Natural Logarithm

The natural logarithm, represented as "ln," is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. In mathematics,

• It is used extensively to model exponential growth and decay.
• "ln(x)" compresses values between 0 and 1 and stretches those greater than 1.
• It is an inverse operation to exponential functions of base \( e \).

In the given solution, taking the natural logarithm \( \ln(\cos(0.5555)) \) helps transition from the geometric significance of cosine to an algebraic manipulation that can easily interact with trigonometric functions. Knowing the properties of the natural logarithm is crucial for handling complex expressions, creating a seamless flow from one function to the next.

Calculator Approximation

Calculator approximation is a technique to obtain close estimates of mathematical expressions. Using a calculator, especially for functions like sine, cosine, and natural logarithm, can

• Speed up computations.
• Provide accurate results with high precision.
• Allow dealing with complex expressions without extensive manual calculations.

By using a calculator to compute each function, starting from \( \cos(0.5555) \) and progressing through \( \ln(x) \) to \( \sin(x) \), and finally squaring the sine result, you ensure calculations are efficient and reliable. This methodological use of approximation is essential in educational contexts, as it reinforces students’ problem-solving strategies while leveraging technology.