91Ó°ÊÓ

When dealing with trigonometric functions, it’s essential to understand how to find a reference angle. A reference angle is the smallest angle formed by the terminal side of a given angle and the x-axis in standard position.
For any given angle, we can calculate the reference angle as follows:

• If the angle is in the first quadrant (0° to 90°), the reference angle is the angle itself.
• If the angle is in the second quadrant (90° to 180°), subtract the angle from 180°.
• If the angle is in the third quadrant (180° to 270°), subtract 180° from the angle.
• If the angle is in the fourth quadrant (270° to 360°), subtract the angle from 360°.

In our example, the initial angle was 1050°. After converting it to standard position, we got 330°. Since 330° is in the fourth quadrant, the reference angle is found using 360° - 330° = 30°.

The sine function is one of the primary trigonometric functions, closely related to the geometry of circles. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. Using symbols, we say:

\(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)

For unit circles, the sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the circle.
In our exercise, the sine of 330° was found by determining it has the same sine value as its reference angle, 30°. But since 330° is in the fourth quadrant, sine is negative, making:\(\sin 330^\circ = -\sin 30^\circ = -\frac{1}{2}\).
Therefore, the sine of 1050° which maps to 330° in standard position is -1/2.

An angle in standard position is one whose vertex is at the origin of the coordinate system, and its initial side lies along the positive x-axis. Here's how to handle angles in standard position:

• If the angle is between 0° and 360°, it is already in standard position.
• If the angle is greater than 360°, subtract multiples of 360° until the angle is within 0° to 360°.
• If the angle is negative, add multiples of 360° until the angle is within 0° to 360°.

In the exercise, 1050° was reduced to 330° by repeatedly subtracting 360° until the angle was within 0° to 360°. This technique can be applied to any angle to convert it to standard position in a straightforward manner.