/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 Find the approximate value of ea... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 32

Find the approximate value of each expression. Round to four decimal places. $$\tan (1.6)$$

Short Answer

Expert verified
The approximate value of \(\tan(1.6)\) is 29.0914.

Step by step solution

01

- Understanding the Tangent Function

The tangent function \(\tan(x)\) is a trigonometric function that represents the ratio of the opposite side to the adjacent side of a right triangle. It can also be defined using the sine and cosine functions: \(\tan(x) = \frac{\text{sin}(x)}{\text{cos}(x)}\). For this exercise, we need to find an approximation of \(\tan(1.6)\).
02

- Use a Calculator

To find the approximate value of \(\tan(1.6)\), use a scientific calculator. Ensure the calculator is in radian mode, as 1.6 is given in radians.
03

- Calculate and Round

Type \(\tan(1.6)\) into the calculator. The result is approximately 29.0913838. Round this value to four decimal places to get 29.0914.

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Key Concepts

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

trigonometric functions
Trigonometric functions are fundamental in understanding the geometry of triangles and the behavior of waves. One essential trigonometric function is the tangent function, denoted as \(\tan(x)\). This function measures the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. Another way to express \(\tan(x)\) is by using sine and cosine: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). So, if you know the values of sine and cosine, you can easily find the tangent. These functions are crucial in fields like engineering, physics, and even in navigation. They help understand periodic phenomena such as sound and light waves.
rounding numbers
Rounding numbers is a basic math skill that helps simplify figures when precision beyond a certain point isn't needed. In many science and engineering contexts, we round the results to make them more readable and practical. For instance, \(\tan(1.6) = 29.0913838\), which can be rounded to four decimal places for simplicity: 29.0914. The general rule for rounding is: if the next digit is 5 or higher, round up; if it's 4 or lower, round down. This step ensures that calculations remain manageable while still accurate enough for practical use.
right triangle ratios
Right triangle ratios are a foundation in trigonometry. They describe the relationships between the sides and angles of a right triangle. These ratios include sine, cosine, and tangent. For the tangent function, it's the ratio of the length of the opposite side to the adjacent side. If you have a right triangle where one angle is \(1.6 \text{radians}\), the ratio of the opposite side to the adjacent side will give you \(\tan(1.6)\). Memorizing these ratios is very useful for solving various geometric and trigonometric problems.
scientific calculator usage
Using a scientific calculator correctly is crucial for solving trigonometric problems. Begin by ensuring that your calculator is set to the correct mode, either degrees or radians, depending on the given angle. For the expression \(\tan(1.6)\), set your calculator to radian mode, as 1.6 is in radians. Then:
  • Type in \(\tan(1.6)\)
  • Press the 'Enter' or '=' key
  • You should get a result similar to 29.0913838
Finally, round this number to the required decimal places, in this case, four, to get 29.0914. These steps might seem simple, but accuracy in mode setting and rounding are crucial in obtaining precise results.

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