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Chapter 1: Problem 24

\(t=\frac{5 \pi}{6}\)

Short Answer

Expert verified
Following the above calculation, you'll find that the approximate value of \( t \) is 2.618

Step by step solution

01

Evaluate the Expression

The value of \( t \) can be found by simply performing the division operation. Substitute the approximate value of \( \pi \) in the equation to get: \( t=\frac{5 \times 3.14159}{6} \)

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Key Concepts

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

Pi Approximation
Pi (\( \pi \)) is a special number in mathematics representing the ratio of the circumference of any circle to its diameter. While its exact value goes on forever without repeating, we often use approximations for simplicity. The most common approximations are 3.14 or 3.14159.

Using these approximations helps us in calculations without using the full, infinite decimal. For example, in the expression \( t=\frac{5 \pi}{6} \), substituting \( \pi \) with 3.14159 allows us to calculate a numerical value for \( t \)efficiently. This is crucial for simplifying complex problems where precise results are not mandatory.
Angle Measurement
Angles are a way to express the rotation between two intersecting lines or rays. They can be measured in degrees or radians. Understanding angle measurement is crucial in geometry and trigonometry.

In expressions like \( t=\frac{5 \pi}{6} \), the angle can represent a fraction of the circle. The circle is divided into units, and angles are measured based on this division. It is vital to know how to translate between different angle measurements, like from degrees to radians, to solve various mathematical problems.
Radians
Radians are a unit of angular measurement used in many areas of mathematics. One full rotation around a circle is \( 2\pi \)radians, which is equivalent to 360 degrees.

This measurement is often favored in mathematics because it simplifies many calculations. For instance, converting radians into a numerical value, such as in \( t=\frac{5 \pi}{6} \), allows for straightforward trigonometric calculations. Recognizing that \( \pi \approx 3.14159 \), makes it easier to work with radians practically.
  • A half-circle is \( \pi \)or 180 degrees.
  • A quarter circle is \( \frac{\pi}{2} \)or 90 degrees.
Understanding radians helps to simplify mathematical operations involving angles.
Mathematical Operations
Mathematical operations involve basic calculations like addition, subtraction, multiplication, and division. These operations form the foundation of solving equations.

In the expression \( t=\frac{5 \pi}{6} \), the operation involves multiplying 5 by an approximation of \( \pi \), and then dividing by 6.

Breaking it down helps:
  • First, perform the multiplication \( 5 \times 3.14159 \).
  • Next, divide the result by 6.
Understanding these sequential steps makes it easier to grasp how to evaluate similar expressions and improve problem-solving skills in mathematics.

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Most popular questions from this chapter

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