## sin210º

### To find the value of sin 210º, we need to understand that sine is a trigonometric function that gives the ratio of the length of the opposite side of a right triangle to the length of the hypotenuse

To find the value of sin 210º, we need to understand that sine is a trigonometric function that gives the ratio of the length of the opposite side of a right triangle to the length of the hypotenuse. However, 210º is not an angle that corresponds to a standard right triangle in the first quadrant.

To solve this problem, we can use the concept of reference angles. The reference angle is the acute angle between the terminal side of an angle in standard position and the x-axis. In this case, the angle 210º is in the third quadrant, but its reference angle can be found in the first quadrant.

180º is the closest angle to 210º in the third quadrant. To find the reference angle, we subtract the given angle from 180º:

Reference angle = 180º – 210º = -30º

Now, we have a reference angle of -30º, which can be converted to a positive angle in the first quadrant by adding 360º:

Positive reference angle = -30º + 360º = 330º

Since the sine function is positive in the second and fourth quadrants, where the angle is measured from the x-axis, we will take the sine of the positive reference angle:

sin 330º

To proceed, we need to find the exact value of sin 330º. We can use the unit circle to determine the sine of this angle. The unit circle is a circle with a radius of 1, where the x-coordinate represents the cosine of an angle, and the y-coordinate represents the sine of an angle.

To find the value of sin 330º, we locate the angle 330º on the unit circle. The terminal side intersects the unit circle at a point (-√3/2, -1/2). The y-coordinate at that point is -1/2, which is equal to sin 330º.

Hence, sin 210º = sin 330º = -1/2.

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