What is the value of sin 120?

What is the value of sin 120?
The value of sin 120 degrees is √3/2.

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. One of the most commonly used trigonometric functions is sine, denoted as sin. The sine of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the hypotenuse.

The question arises, what is the value of sin 120? To find the answer, we need to understand the concept of the unit circle. The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate system.

When an angle is measured in degrees, as in this case, the positive x-axis is considered as 0 degrees, and angles are measured counterclockwise. So, if this unit circle is divided into 360 degrees, we can find the value of sin 120 by determining the y-coordinate of the point on the unit circle that corresponds to 120 degrees.

To calculate sin 120, we draw a line from the origin to the point P on the unit circle that forms a 120-degree angle with the positive x-axis. This line segment divides the unit circle into two equal parts. The y-coordinate of point P represents the sine value for the angle 120 degrees.

Now, let’s find the value of sin 120. By observing the unit circle, we can see that point P lies in the second quadrant. The y-coordinate in this quadrant is positive, and the x-coordinate is negative. To determine the length of the y-coordinate, we calculate the perpendicular distance from point P to the x-axis.

Since the unit circle has a radius of 1 unit, the perpendicular distance from point P to the x-axis is equal to the altitude of an equilateral triangle with a side length of 1 unit. By using the Pythagorean theorem, we can find this altitude as follows:

(Side length)^2 = (Altitude)^2 + (1/2 * Side length)^2
1^2 = (Altitude)^2 + (1/2)^2
(Altitude)^2 = 1 – 1/4
(Altitude)^2 = 3/4
Altitude = √3/2

Therefore, the y-coordinate, which is equal to the value of sin 120, is √3/2. **The value of sin 120 is √3/2.** It signifies that the sine of 120 degrees is positive and equal to √3/2.

FAQs:

1. What does sin mean in trigonometry?

Sin (short for sine) is one of the basic trigonometric functions that represents the ratio of the length of the side opposite to an angle in a right-angled triangle to the hypotenuse.

2. How do you calculate the sine of an angle?

The sine of an angle can be calculated by dividing the length of the side opposite the angle by the length of the hypotenuse in a right-angled triangle.

3. What is the unit circle?

The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate system, used to visualize trigonometric functions and their values.

4. How many degrees are there in a circle?

There are 360 degrees in a complete circle.

5. How is the unit circle divided?

The unit circle is divided into 360 degrees, with the positive x-axis as 0 degrees, and angles are measured counterclockwise.

6. Which quadrant does sin 120 belong to?

Sin 120 belongs to the second quadrant of the unit circle.

7. What is the range of the sine function?

The range of the sine function is between -1 and 1, inclusive.

8. What is the value of sin 30?

The value of sin 30 degrees is 1/2.

9. What is the value of sin 45?

The value of sin 45 degrees is √2/2.

10. What is the value of sin 60?

The value of sin 60 degrees is √3/2.

11. What is the value of sin 90?

The value of sin 90 degrees is 1.

12. Can the sine of an angle be larger than 1?

No, the sine of an angle is always between -1 and 1, inclusive.

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