Find the indicated function value. If it is undefined, say so. See Example 4. sin 90°

Verified step by step guidance

1. Recall that the sine function, \(\sin \theta\), gives the ratio of the length of the side opposite the angle \(\theta\) to the hypotenuse in a right triangle, or equivalently, the y-coordinate of the point on the unit circle at angle \(\theta\).
2. Identify the angle given: here, the angle is \(90^\circ\), which corresponds to the point on the unit circle at the top of the circle.
3. On the unit circle, the coordinates at \(90^\circ\) are \((0, 1)\), where the x-coordinate is \(\cos 90^\circ\) and the y-coordinate is \(\sin 90^\circ\).
4. Therefore, \(\sin 90^\circ\) is equal to the y-coordinate of this point, which is 1.
5. Conclude that \(\sin 90^\circ\) is defined and its value is 1.

Key Concepts

Definition of the Sine Function

The sine function relates an angle in a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse. It is also defined on the unit circle as the y-coordinate of the point corresponding to the angle measured from the positive x-axis.

Special Angles and Their Sine Values

Certain angles, like 0°, 30°, 45°, 60°, and 90°, have well-known sine values that are often memorized. For example, sin 90° equals 1 because at 90°, the point on the unit circle is at (0,1), making the sine value the maximum possible.

Domain and Range of the Sine Function

The sine function is defined for all real numbers (angles), so sin 90° is defined. Its range is between -1 and 1, inclusive, meaning sine values cannot be greater than 1 or less than -1.