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Chapter 6: Problem 74

In Exercises \(71-76,\) find all the solutions of the equation. $$\sin t=-1$$

Short Answer

Expert verified
Question: Determine all the values of \(t\) for which the equation \(\sin t = -1\). Answer: The general solution for all values of \(t\) for which \(\sin t = -1\) is given by the equation: $$t = \frac{3\pi}{2} + 2\pi k$$, where \(k\) is an integer.

Step by step solution

01

Find the basic solution

Recall that \(\sin t = -1\) when \(t\) is an odd multiple of \(\frac{\pi}{2}\). In the range \(0 鈮� t < 2\pi\), there is only one such value: $$t = \frac{3\pi}{2}$$
02

Generalize the solution

The sine function is periodic with period \(2\pi\). This means that any angle that is a multiple of \(2\pi\) added to the basic solution will also be a solution for this equation. So, we can write the general solution for all \(t\) as: $$t = \frac{3\pi}{2} + 2\pi k$$ where \(k\) is an integer. This equation gives all the values of \(t\) for which \(\sin t = -1\).

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Key Concepts

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

Sine Function
When tackling problems involving sine functions, it's important to grasp what the sine function represents. The sine function is a fundamental trigonometric function often used in mathematics to describe oscillations or waves. It relates the angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse:
  • The sine of an angle \( t \) is denoted as \( \sin t \).
  • The range of the sine function is from -1 to 1, denoting its amplitude.
In circular terms, sine can also help determine vertical positions on the unit circle. For our specific problem, knowing that \( \sin t = -1 \) means we are identifying the angle \( t \) at which the sine wave reaches its minimum point.
Periodic Functions
Periodic functions have a repeating pattern over a set interval. The trigonometric sine function is a classic example of a periodic function, having a period of \(2\pi\). This means the sine wave repeats every \(2\pi\) radians, quite like a cycle.
  • A key feature of periodic functions is their ability to predict future values based on understanding one period.
  • For sine, these periodicity characteristics allow us to find all possible solutions to \( \sin t = -1 \) through generalization.
Thus, by understanding its periodic nature, once we find a basic solution such as \( \frac{3\pi}{2} \), we can generate additional solutions by adjusting by \(2\pi\), yielding a series of angles that fulfill the original sine condition.
General Solutions
The concept of general solutions in trigonometry involves expressing infinite solutions of equations like \( \sin t = -1 \) in a concise form. By utilizing the natural characteristics of trigonometric functions, especially their periodicity, we derive general solutions that encompass all possible angles that satisfy the equation.
  • For the sine function, given that it repeats every \(2\pi\), our general solution format is based on adding multiples of \(2\pi\) to the basic solutions.
  • For example, starting from \( t = \frac{3\pi}{2} \), the general solution is \( t = \frac{3\pi}{2} + 2\pi k \), where \( k \) represents any integer.
This general solution provides a comprehensive set of answers, showing that every angle of the form \( \frac{3\pi}{2} + 2\pi k \) meets the equation \( \sin t = -1 \).

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Most popular questions from this chapter

Sketch a complete graph of the function. $$q(t)=\frac{2}{3} \cos \frac{3}{2} t$$

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