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Chapter 1: Problem 78

Use a calculator to find the value of each function. Round answers to four decimal places. $$ \cot (\pi / 10) $$

Short Answer

Expert verified
\(\text{cot}(\frac{\pi}{10}) \approx 3.0777\)

Step by step solution

01

- Understand the Cotangent Function

The cotangent function, \(\text{cot}(x)\), is defined as the reciprocal of the tangent function. Therefore, \(\text{cot}(x) = \frac{1}{\text{tan}(x)}\).
02

- Convert Radians to Degrees (if needed)

Many calculators have a mode for radians and degrees. Ensure your calculator is set to radians because \(\frac{\pi}{10}\) is in radians.
03

- Calculate the Tangent

Use the calculator to find \(\tan(\frac{\pi}{10})\). Input \(\frac{\pi}{10}\) into the tangent function of the calculator to get the value.
04

- Find the Cotangent

Since \(\text{cot}(\frac{\pi}{10}) = \frac{1}{\tan(\frac{\pi}{10})}\), take the reciprocal of the tangent value obtained in the previous step.
05

- Round the Answer

Round the result to four decimal places for the final answer.

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Key Concepts

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

cotangent function
The cotangent function, denoted as \(\text{cot}(x)\), is a fundamental trigonometric function. It represents the reciprocal of the tangent function. That is, \( \text{cot}(x) = \frac{1}{\tan(x)} \). If you know the value of the tangent of an angle, you can easily find the cotangent by simply taking its reciprocal.
The cotangent function is particularly useful in various mathematical contexts, including coordinate geometry, trigonometric identities, and calculus.
In summary:
  • The cotangent function is the reciprocal of the tangent function.
  • If \( \tan(x) = a \), then \( \text{cot}(x) = \frac{1}{a} \).
tangent function
The tangent function, represented as \(\text{tan}(x)\), is another key trigonometric function. It is defined as the ratio of the sine to the cosine of an angle: \( \tan(x) = \frac{\text{sin}(x)}{\text{cos}(x)} \). This function helps in solving triangles and modeling periodic phenomena.
To use a calculator to find \(\text{tan}(x)\), input the angle in the proper format (degrees or radians). Make sure your calculator is set to the correct mode. For example, if your angle is given in radians, the calculator should be in radian mode.
Key points to remember:
  • The tangent of an angle is the sine of that angle divided by the cosine of that angle.
  • Ensure that your calculator is in the correct mode (radians or degrees) before calculating.
radian measure
Angles can be measured in degrees or radians. The radian measure is one of the most commonly used measures in higher mathematics. One radian is equivalent to the angle created when the radius is wrapped around the circle's circumference. In simplest terms, \( \text{1 radian} = \frac{180}{\text{pi}} \) degrees.
In trigonometry, radians provide a natural way of measuring angles that simplifies many mathematical equations and formulas. This is crucial when working with angles in calculus and other advanced mathematics.
Practical tips:
  • When using trigonometric functions in higher math, ensure you are comfortable converting between degrees and radians.
  • Remember, \( \text{pi} \) radians equals 180 degrees. Hence, \( \frac{\text{pi}}{10} \) radians is \( 18 \) degrees.
reciprocal function
The concept of a reciprocal function is essential in mathematics. The reciprocal of a function \( f(x) \) is \( \frac{1}{f(x)} \). For trigonometric functions, knowing how to find reciprocals can simplify complex equations.
For example, the reciprocal of the tangent function is the cotangent function: if \( \text{tan}(x) = a \), then \( \text{cot}(x) = \frac{1}{a} \). This principle helps find values quickly and efficiently.
Main takeaways:
  • A reciprocal flips the numerator and denominator of a fraction.
  • For trigonometric functions, knowing the reciprocals helps in problem-solving and simplification.

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