Problem 221 - Precalculus

Question

For the following exercises, use reference angles to evaluate the expression. If \(\sin t=\frac{\sqrt{2}}{2}, \quad\) what is the \(\sin (-t) ?\)

Step-by-Step Solution

Answer

\( \sin(-t) = -\frac{\sqrt{2}}{2} \)

Step 1: Understand the Given Information

We are given that \( \sin t = \frac{\sqrt{2}}{2} \). This value corresponds to a well-known angle in the unit circle. Recall that \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). Thus, \( t \) could correspond to the angle \( \frac{\pi}{4} \) or any other angle with this sine value in other quadrants.

Step 2: Consider the Property of Sine Function

The sine function is odd, meaning that \( \sin(-x) = -\sin(x) \) for any angle \( x \). Therefore, \( \sin(-t) = -\sin(t) \).

Step 3: Apply the Property to Find \( \sin(-t) \)

Using the odd function property of sine, substitute the value of \( \sin t \): \( \sin(-t) = -\sin(t) = -\frac{\sqrt{2}}{2} \).

Step 4: Interpret the Result

The result \( \sin(-t) = -\frac{\sqrt{2}}{2} \) indicates that the sine of \( -t \) is simply the negative of the sine of \( t \), due to the odd symmetry of the sine function.

Key Concepts

Reference Angles

A reference angle is the smallest angle that a given angle makes with the x-axis. It is always a positive angle and is used to simplify the process of finding trigonometric values of any angle. The idea behind reference angles helps us to work within the familiar 0 to \( \frac{\pi}{2} \) (or 0 to 90 degrees) range, making calculations easier.
Consider this:

• To find a reference angle, determine how far the given angle is from the nearest x-axis.
• If the angle is in the first quadrant, the reference angle is the angle itself.
• In the second quadrant, subtract the angle from \( \pi \).
• In the third quadrant, subtract \( \pi \) from the angle.
• In the fourth quadrant, subtract the angle from \( 2\pi \).

Understanding reference angles aids in evaluating trigonometric functions, as angles that differ by a full rotation or share reference angles will have relationships in their sine, cosine, and tangent values.

Unit Circle

The unit circle is a crucial concept in trigonometry and calculus. It is a circle in the coordinate plane with a center at the origin (0,0) and a radius of 1. This makes it a perfect tool for understanding how the trigonometric functions work, especially sine and cosine.
On the unit circle:

• The x-coordinate of a point gives the cosine of the angle.
• The y-coordinate gives the sine of the angle.
• A positive rotation moves counterclockwise from the positive x-axis, while a negative rotation goes clockwise.

For example, when \( \, t = \frac{\pi}{4} \), which translates to 45 degrees, the coordinates are \( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \), representing \( \cos \) and \( \sin \).
Recognizing patterns on the unit circle allows us to solve trigonometric problems efficiently and provides a visual way to understand angles and their reference angles.

Odd Functions

Odd functions are special types of functions in mathematics where \( f(-x) = -f(x) \) holds true for all x in the function's domain. This property manifests as symmetry about the origin on the graph.
The sine function, \( \sin(x) \), is a classic example of an odd function. It means that for every pair \( (t, -t) \), the values of \( \sin(t) \) and \( \sin(-t) \) are opposites; for instance, if \( \sin t = \frac{\sqrt{2}}{2} \), then \( \sin(-t) = -\frac{\sqrt{2}}{2} \).
This symmetry simplifies calculations, especially in solving trigonometric identities and equations, because flipping the sign of the angle simply flips the sign of the sine value.