Problem 74 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

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 1: Find the basic solution

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

Step 2: 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 \(\sin t = -1\).

Key Concepts

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 \( \sin t = -1 \).