Question

Use a calculator to approximate each value. $$ \sin ^{2}(\ln (\cos 0.5555)) $$

Short answer

The approximate value of \( \sin^2(\ln(\cos(0.5555))) \) is 0.0263.

Step-by-step solution

Calculate the Cosine

First, we need to find the value of \( \cos(0.5555) \). Use a calculator to find: \[ \cos(0.5555) \approx 0.8490 \]

Find the Natural Logarithm

Next, calculate the natural logarithm (ln) of the value obtained from Step 1. \[ \ln(0.8490) \approx -0.1634 \]

Calculate the Sine

Now, find the sine of the result from Step 2. Use the calculator to find: \[ \sin(-0.1634) \approx -0.1623 \]

Square the Sine Value

Finally, square the sine value obtained in Step 3 to find \( \sin^2(\ln(\cos(0.5555))) \). Calculate: \[ (-0.1623)^2 = 0.0263 \]

Key concepts

Natural Logarithm

The natural logarithm, often abbreviated as ln, is an essential concept in mathematics. It is the logarithm to the base of the mathematical constant \( e \), which is approximately 2.71828. The natural logarithm is denoted as \( \ln(x) \) and is commonly used in calculus and mathematical modeling.

Using a natural logarithm simplifies many mathematical expressions, especially those involving exponential growth or decay. For instance, when calculating interest or population growth, the base \( e \) naturally fits scenarios where the rate of change is proportional to the current amount.

• The natural logarithm is only defined for positive numbers. For example, \( \ln(1) = 0 \) and \( \ln(e) = 1 \).
• It is an inverse function of the exponential \( e^x \); this means \( \ln(e^x) = x \) for all \( x \).

Understanding how to use natural logarithms will help you in interpreting results whenever they appear in equations involving growth rates or changes over time. It's also essential for the calculation of more complex mathematical expressions, as demonstrated in logarithmic identities and properties.

Calculator Usage in Mathematics

Using a calculator is a vital skill when solving math problems, especially when dealing with complex functions like sine, cosine, and logarithms. Here are some tips for effective calculator usage:

Always ensure your calculator is set to the correct mode, such as radians or degrees, depending on the problem at hand.

• For the exercise above, set your calculator to radians because angles are often calculated in radians in advanced mathematics.
• When approximating values such as \( \cos(0.5555) \), ensure that your input is accurate and double-check the results.
• Use the calculator functions wisely; knowing when to apply each function, like \( \sin \) and \( \ln \), will lead you to the correct result.

Regular use of calculators helps develop a more intuitive understanding of how trigonometric and logarithmic functions behave. Larger computational tasks become manageable, and accuracy in calculations can greatly improve with practice.

Step-by-Step Problem Solving

Approaching a problem step-by-step is an effective technique in mathematics that ensures a thorough understanding and accurate solution. The outlined steps not only make complex problems manageable but also build a mindset of logical progression.

Here is a structured approach, similar to the exercise solution provided:

• Break Down the Problem: Identify the different parts of the problem. For example, in the exercise, recognize components like \( \cos(x) \) and \( \ln(y) \).
• Compute Step-by-Step: Calculate each part individually. This focused approach minimizes mistakes and allows for validation at each stage. For instance, assess first the value of cosine before progressing to the logarithm.
• Verify Results: After solving each component, verify the calculations to ensure accuracy. Errors can accumulate, so double-checking can prevent them from magnifying.

This method encourages patience and logical thinking, reinforcing comprehension in complicated mathematical situations. By breaking down exercises into smaller tasks, students can tackle even the most challenging problems with confidence and clarity.