Why is Value for sin 120 Same as sin 30?

The value of sin 120° is the same as sin 30° because they are both reference angles in the unit circle, and they share the same sine value due to their symmetrical positions.

In trigonometry, the sine function (sin) relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. The unit circle, a fundamental concept in trigonometry, helps in understanding why sin 120° equals sin 30°.

The angle of 120° is in the second quadrant of the unit circle, where the sine values are positive. The reference angle for 120° is 60°, which is the acute angle formed with the horizontal axis.

The sine of an angle in the second quadrant is equal to the sine of its reference angle in the first quadrant. Therefore,

sin 120° = sin (180° – 120°) = sin 60°.

Similarly, the angle of 30° is in the first quadrant, and its sine value is positive. Since sin 60° and sin 30° are complementary angles (adding up to 90°), they have the same sine value but with different ratios. Thus, sin 120° = sin 60° = sin 30°.

This concept is widely searched with keywords like “sin 120 degrees”, “sin 120 exact value”, and “sin 120 in fraction”. Understanding these relationships is crucial in fields like mathematics, physics, and engineering, where trigonometric functions are used to model and solve real-world problems.