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Chapter 3: Problem 72

Use a calculator to approximate each value to four decimal places. $$ \cos (-2.5) $$

Short Answer

Expert verified
\(\cos(-2.5) \approx -0.8011\) if calculated in radians.

Step by step solution

01

Understand the Cosine Function

The cosine function, denoted as \(\cos(x)\), is an even function. This means that \(\cos(x) = \cos(-x)\). Therefore, the negative sign in the input does not change the output of the cosine function.
02

Calculate \(\cos(2.5)\)

Use a calculator and input \(2.5\) to find \(\cos(2.5)\). Ensure your calculator is set to the correct mode (radians or degrees) based on the input value.
03

Approximate to Four Decimal Places

The calculator will give you a result which you need to round to four decimal places to meet the requirement of the exercise. If necessary, review how to round numbers correctly to ensure precision.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Even Function
Among the fundamental properties of trigonometric functions, the concept of an even function is critical. An even function has the property that its graph is symmetrical with respect to the y-axis. This means for any function that is even, if you evaluate the function at \(x\) and at \(-x\), you get the same value: \(f(x) = f(-x)\). The cosine function is a classic example of this. This implies when you need to calculate \( \cos(-2.5) \), you can simplify it to \( \cos(2.5) \) because of its even nature.
  • The symmetry of even functions can simplify calculations significantly.
  • Recognizing even functions helps in understanding transform properties of these functions.
  • Visualizing the function as symmetric around the y-axis can provide a graphical advantage in comprehending this property.
Calculator Use in Trigonometry
Using a calculator effectively in trigonometry is an important skill for tackling exercises accurately. When working with trigonometric functions like cosine, your calculator must be set to the correct mode based on the unit of your angle, which might be in degrees or radians. If given an angle in radians, make sure to switch your calculator to radian mode. This ensures the result is accurate according to the context of the problem. For example, when tasked with finding \( \cos(2.5) \) in the context of radians, entering this directly into the calculator in radian mode will give you the precise value.
Here's how to ensure precision:
  • Double-check your calculator mode before inputting numbers.
  • Know the unit of your input value; radians and degrees can lead to very different answers if misconstrued.
  • Practice using your calculator to become familiar with its functions specific to trigonometry.
Rounding Numbers in Mathematics
Once you have the solution from your calculator, it is often necessary to round your result to a certain number of decimal places. In many trigonometry problems, results need to be expressed to four decimal places for precision.
Understanding the rules of rounding is crucial:
  • Observe the digit at the fifth decimal place to determine if rounding is needed. If it is 5 or greater, increase the fourth decimal place by one.
  • For digits less than 5, leave the fourth decimal place as it is.
  • Perform a quick review of rounding principles to make sure you apply them correctly in various contexts.
Practicing these rounding skills is essential, especially for exams where precision could impact your score. Efficient rounding ensures that your answer is not only correct but presented in the format typically required by academic standards.

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