Problem 39 - Contemporary Precalculus

Question

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\sin (-t)=-\sin t$$

Step-by-Step Solution

Answer

Answer: Yes, the equation $$\sin(-t) = -\sin t$$ is an identity, as the graphs of \(\sin (-t)\) and \(-\sin t\) are identical, and the property of sine function being odd confirms this relationship.

Step 1: Understand the problem

We need to determine whether the given equation is an identity or not using graphs. An identity would mean that the equation holds true for all values of t.

Step 2: Graph the left side

To start, graph the left side of the equation: \(\sin(-t)\). Use an online graphing tool or graphing calculator to visualize the graph. You will notice that it is an even function with maximum and minimum values of 1 and -1, respectively.

Step 3: Graph the right side

Next, graph the right side of the equation: \(-\sin(t)\). The graph will also be limited to the interval [-1, 1], but this time it is an odd function.

Step 4: Compare the graphs

Now, compare the graphs of both sides of the equation. If the graphs are the same, then the equation is an identity. In this case, you will notice that the graphs are identical, meaning that the equation is true for all values of t.

Step 5: Conclusion

Since the graphs of \(\sin (-t)\) and \(-\sin t\) are identical, the equation $$\sin(-t) = -\sin t$$ is indeed an identity. It holds true for all values of t as the trigonometry property states that sine function is odd, which means \(\sin(-t) = -\sin t\).

Key Concepts

Understanding the Sine Function

The sine function is one of the primary trigonometric functions and plays a critical role in expressing real-world phenomena such as waves and oscillations.

At its core, the sine function can be thought of as a way to measure the vertical position of a point rotating around a unit circle. More formally, if we consider an angle \( t \), measured in radians, the sine of \( t \), denoted as \( \text{sin}(t) \), is the y-coordinate of the point on the unit circle at this angle from the positive x-axis.

Key Properties of the Sine Function

• It is periodic with a period of \( 2\text{pi} \) radians.
• The range of the sine function is from -1 to 1.
• It is an odd function, meaning \( \text{sin}(-t) = -\text{sin}(t) \).

Understanding the sine function is essential for solving trigonometric problems and modeling periodic behavior.

Odd and Even Functions

In mathematics, the concept of odd and even functions helps to understand function symmetry about the origin or the y-axis. For any real function \( f(x) \), the function can either be:

Even Function

If \( f(-x) = f(x) \) for all values of \( x \), the function's graph is symmetrical about the y-axis. A typical example is the cosine function, \( \text{cos}(x) \).

Odd Function

If \( f(-x) = -f(x) \) for all values of \( x \), the graph of the function is symmetrical about the origin. An example of an odd function is the sine function, \( \text{sin}(x) \), which relates directly to the exercise at hand where \( \text{sin}(-t) = -\text{sin}(t) \) establishes its odd nature.

Recognizing whether a function is odd or even can greatly simplify solving equations, analyzing graphs, and predicting function behavior.

Graphing Trigonometric Functions

Graphing trigonometric functions such as sine, cosine, and tangent can provide a visual representation of their behavior over various intervals. The process often includes identifying key properties like amplitude, period, phase shifts, and reflections.

For the sine function:\( \text{sin}(x) \), a graph can be sketched by plotting points from the value of \( x \) on the x-axis and the corresponding sine value on the y-axis. Typically, the graph will start from the origin and move in a smooth, continuous wave-like pattern that repeats every \( 2\text{pi} \) radians.

The exercise shows the graph of \( \text{sin}(-t) \) and its equivalent, \( -\text{sin}(t) \), allowing us to visually confirm an important trigonometric identity. By graphing, students reinforce their understanding and gain valuable skills in interpreting the periodic nature of these functions. This graphical approach is not only helpful in verifying identities but also in learning to predict and analyze the behavior of trigonometric functions in applied contexts.