The unit circle has been divided into twelve equal arcs, corresponding to t-values of
0, 𝜋/6, 𝜋/3, 𝜋/2, 2𝜋/3, 5𝜋/6, 𝜋, 7𝜋/6, 4𝜋/3, 3𝜋/2, 5𝜋/3, 11𝜋/6, and 2𝜋


Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
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sin 3𝜋/2

Verified step by step guidance

1. Recall that on the unit circle, the coordinates of a point corresponding to an angle \(t\) are given by \((\cos t, \sin t)\).
2. Identify the angle \(t = 3\pi/2\) on the unit circle. This angle corresponds to rotating \(270^\circ\) counterclockwise from the positive x-axis.
3. Find the coordinates of the point on the unit circle at \(t = 3\pi/2\). This point lies on the negative y-axis.
4. Since the sine of an angle is the y-coordinate of the corresponding point on the unit circle, write \(\sin 3\pi/2\) as the y-coordinate of that point.
5. State the value of \(\sin 3\pi/2\) based on the coordinates found, or note if it is undefined (which it is not in this case).

Key Concepts

Unit Circle and Radian Measure

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles on the unit circle are measured in radians, where 2π radians correspond to a full rotation of 360°. Dividing the circle into twelve equal arcs means each arc measures π/6 radians, providing standard angle measures for evaluating trigonometric functions.

Coordinates on the Unit Circle and Trigonometric Functions

Each point on the unit circle corresponds to an angle t and has coordinates (x, y), where x = cos(t) and y = sin(t). These coordinates allow direct evaluation of sine and cosine values for given angles. For example, sin(3π/2) corresponds to the y-coordinate of the point at angle 3π/2.

Definition and Domain of Trigonometric Functions

Sine and cosine functions are defined for all real numbers and correspond to y and x coordinates on the unit circle, respectively. Other functions like tangent are defined as sin(t)/cos(t) and may be undefined where cosine is zero. Understanding when functions are defined or undefined is crucial for correctly evaluating expressions at specific angles.