Problem 71
Question
Evaluate to four significant digits. $$\sin \frac{\pi}{8} \tan \frac{\pi}{8}$$
Step-by-Step Solution
Verified Answer
\(\sin \frac{\pi}{8} \tan \frac{\pi}{8} \approx 0.3827\)
1Step 1: Calculate \(\sin \frac{\pi}{8}\)
To find the value of \(\sin \frac{\pi}{8}\), use a calculator or trigonometric tables. Since the sine function is not standard for \( \frac{\pi}{8} \) radians, a calculator will likely be necessary.
2Step 2: Calculate \(\tan \frac{\pi}{8}\)
To find the value of \(\tan \frac{\pi}{8}\), use a calculator or trigonometric tables. Since the tangent function is also not standard for \( \frac{\pi}{8} \) radians, a calculator or an identity involving half angles might be necessary to evaluate it.
3Step 3: Multiply the Results
Multiply the results from Step 1 and Step 2 to find \(\sin \frac{\pi}{8} \cdot \tan \frac{\pi}{8}\). Make sure to round it to four significant digits.
Key Concepts
Understanding the Sine FunctionThe Tangent Function ExplainedImportance of Significant Digits
Understanding the Sine Function
The sine function is one of the fundamental concepts in trigonometry, significant in fields such as physics, engineering, and mathematics. It relates the angles of a right triangle to the ratio of the side opposite the angle to the hypotenuse. The sine of any angle is equal to the y-coordinate of that angle on the unit circle.
When calculating \( \sin \frac{\pi}{8} \), a commonly used approach involves a calculator. Since \( \frac{\pi}{8} \) radians is not a commonly memorized angle, a calculator provides the decimal representation. The calculation requires careful attention to ensure the calculator is set to the correct mode (radians or degrees) based on the problem requirements. Trigonometric tables, if available, may also help but are less commonly used in contemporary practice.
When calculating \( \sin \frac{\pi}{8} \), a commonly used approach involves a calculator. Since \( \frac{\pi}{8} \) radians is not a commonly memorized angle, a calculator provides the decimal representation. The calculation requires careful attention to ensure the calculator is set to the correct mode (radians or degrees) based on the problem requirements. Trigonometric tables, if available, may also help but are less commonly used in contemporary practice.
The Tangent Function Explained
The tangent function is another trigonometric function that is crucial for analyzing angles and triangles. It is defined as the ratio of the sine to the cosine of a given angle. In right triangles, it represents the ratio of the side opposite to the adjacent side of the specified angle. Like the sine function, tangent values for standard angles can often be found in trigonometric tables or calculated using a calculator.
When evaluating \( \tan \frac{\pi}{8} \), the use of the half-angle identity might also be a valuable approach. The half-angle identities are trigonometric formulas that express trigonometric functions of half-angles in terms of the functions of the original angle. They are particularly useful when dealing with non-standard angles like \( \frac{\pi}{8} \). For practical purposes, using a calculator is usually the fastest method to find the tangent of angles that are not common in trigonometric identities.
When evaluating \( \tan \frac{\pi}{8} \), the use of the half-angle identity might also be a valuable approach. The half-angle identities are trigonometric formulas that express trigonometric functions of half-angles in terms of the functions of the original angle. They are particularly useful when dealing with non-standard angles like \( \frac{\pi}{8} \). For practical purposes, using a calculator is usually the fastest method to find the tangent of angles that are not common in trigonometric identities.
Importance of Significant Digits
Significant digits, or significant figures, play a vital role in scientific and mathematical calculations by indicating the precision of a measurement. They include all the digits that are known with certainty plus the first digit that is uncertain. The rules of significant digits dictate how to round the numbers and maintain the accuracy of computations.
For the problem at hand, rounding the product of \( \sin \frac{\pi}{8} \) and \( \tan \frac{\pi}{8} \) to four significant digits ensures that the answer reflects the precision intended for the calculation. To adhere to these rules, identify the first four digits with certainty after performing the multiplicative operation and round off any subsequent number based on standard rounding guidelines. Being adept with significant digits is essential, as it affects the reliability of the outcomes in both theoretical and applied contexts.
For the problem at hand, rounding the product of \( \sin \frac{\pi}{8} \) and \( \tan \frac{\pi}{8} \) to four significant digits ensures that the answer reflects the precision intended for the calculation. To adhere to these rules, identify the first four digits with certainty after performing the multiplicative operation and round off any subsequent number based on standard rounding guidelines. Being adept with significant digits is essential, as it affects the reliability of the outcomes in both theoretical and applied contexts.
Other exercises in this chapter
Problem 69
Evaluate to four significant digits. $$\cos ^{2} \frac{3 \pi}{4}$$
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Evaluate to four significant digits. $$\frac{\pi}{6} \sin \frac{\pi}{6}$$
View solution Problem 72
Evaluate to four significant digits. $$3 \sin \frac{\pi}{9} \cos ^{2} \frac{\pi}{9}$$
View solution Problem 73
Find the area of a sector having a radius of 5.92 in. and a central angle of \(62.5^{\circ} .\)
View solution