Find the indicated function value. If it is undefined, say so. See Example 4. sin(―270°)

Verified step by step guidance

1. Recall the definition of the sine function for negative angles: \(\sin(-\theta) = -\sin(\theta)\).
2. Rewrite the given expression using this identity: \(\sin(-270^\circ) = -\sin(270^\circ)\).
3. Determine the value of \(\sin(270^\circ)\) by considering the unit circle. The angle \(270^\circ\) corresponds to the point \((0, -1)\) on the unit circle, so \(\sin(270^\circ) = -1\).
4. Substitute this value back into the expression: \(\sin(-270^\circ) = -(-1)\).
5. Simplify the expression to find the value of \(\sin(-270^\circ)\).

Key Concepts

Understanding the Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles are measured from the positive x-axis, with positive angles rotating counterclockwise and negative angles clockwise. Knowing how to locate angles like -270° on the unit circle helps determine the sine value.

Sine Function Definition on the Unit Circle

The sine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the unit circle. This means sin(θ) equals the vertical position on the circle, which can be positive, negative, or zero depending on the angle's quadrant.

Angle Coterminality and Reference Angles

Angles differing by full rotations (multiples of 360°) share the same terminal side and thus the same sine value. To find sin(-270°), convert it to a coterminal positive angle by adding 360°, simplifying the evaluation using known sine values of standard angles.