$\begin{array}{r}
\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right) \
\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)
\end{array}$
The unit circle shows the sine and cosine values for particular angles created by points along the circle. Each ordered pair along the circle represents a cosine and sine value, respectively.
Trigonometry, whether calculated in triangle or circle form, is a very useful and powerful mathematics system used in both ancient times and today.
Which statement is true regarding the values of sine on the unit circle?
$\sin (\theta)=x$
$\sin (\theta)=y$
$\sin (\theta)=\frac{x}{y}$
$\sin (\theta)=\frac{y}{x}$
Answer (2)
The problem relates to understanding sine values on the unit circle.
The x-coordinate on the unit circle represents cos ( θ ) , and the y-coordinate represents sin ( θ ) .
Therefore, sin ( θ ) = y .
The correct statement is sin ( θ ) = y .
Explanation
Problem Analysis Let's analyze the problem. We are given two points on the unit circle and asked to identify the correct relationship between the sine of an angle and the coordinates of a point on the unit circle.
Unit Circle Definition Recall the definition of the unit circle. A point on the unit circle is defined as ( cos ( θ ) , sin ( θ )) , where θ is the angle formed with the positive x-axis. Therefore, the x-coordinate of the point is the cosine of the angle, and the y-coordinate is the sine of the angle.
Sine and y-coordinate Based on the unit circle definition, we have:
sin ( θ ) = y
This means that the sine of the angle θ is equal to the y-coordinate of the point on the unit circle.
Correct Statement Comparing this with the given options, we can see that the correct statement is:
sin ( θ ) = y
Examples
Understanding the sine value on the unit circle is crucial in various fields. For example, in physics, when studying simple harmonic motion, the sine function describes the position of an object oscillating over time. In engineering, it helps analyze alternating current (AC) circuits, where voltage and current vary sinusoidally. Also, in navigation, sine values are used to determine positions and angles, especially in GPS systems and celestial navigation.
The sine of an angle on the unit circle corresponds to the y-coordinate of the point, which means sin ( θ ) = y . The correct answer is sin ( θ ) = y .
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