Value of Sin 60

sin 60°

Let's explore the value of sin 60 degrees and how to express it in various forms. Sin 60 degrees can be written in radians as sin (60° × π/180°), which simplifies to sin (π/3) or sin (1.0471975...).

Here are the key points:

  • Sin 60° in decimal form: 0.8660254...
  • Sin 60° in fractional form: √3/2
  • Sin (-60°): -0.8660254...
  • Sin 60° in radians: sin (π/3) or sin (1.0471975...)

To better understand, consider the unit circle and trigonometric identities. Sin 60° corresponds to the y-coordinate of a point on the unit circle at a 60-degree angle from the positive x-axis. This value, √3/2, is derived from the properties of a 30-60-90 triangle. Additionally, expressing angles in radians is a common practice in higher mathematics, where 60 degrees equals π/3 radians.

Also Check: Cos Inverse Formula

What is the Value of Sin 60 Degrees?

The sine of 60 degrees has a decimal value of approximately 0.866025403. To convert this angle into radians, we use the degree-to-radian conversion formula: radians = degrees × (π / 180°).

Applying this formula:

60° × (π / 180°) = π / 3 ≈ 1.04719

Thus, 60 degrees is equivalent to approximately 1.04719 radians.

Therefore, sin(60°) = sin(1.04719) ≈ √3 / 2 ≈ 0.8660254.

Explanation:

The decimal value of sin 60 degrees is 0.866025403. The equivalent of the angle (60 degrees) in
radians can likewise be used to express Sin 60 degrees (1.04719 . . .).
Using the degree to radian conversion, we know that in radians = in degrees (pi/180°).
Note: Because sine is an odd function, sin(-60°) value is -sin(-60°).

Methods to Find Sin 60

In the first quadrant, the sine function has positive values. For example, the sine of 60° is approximately 0.866. You can determine the value of sin 60° using different methods, such as:

  • Trigonometric Functions: By applying the properties and definitions of trigonometric functions, specifically for 60°, you can find that sin 60° equals √3/2, which is approximately 0.866.

  • Unit Circle: By examining the unit circle, where the coordinates of points represent the values of sine and cosine for corresponding angles, you can locate the angle 60°. The y-coordinate at this angle corresponds to the sine value, which is √3/2 or approximately 0.866.

These approaches help in understanding why sin 60° is 0.866 and how it is derived from basic trigonometric principles.

Sin 60 Degrees Using Trigonometric Formulas

We can represent sin 60 degrees using various trigonometric formulas as follows:

  • ± √(1 - cos²(60°))
  • ± tan 60°/√(1 + tan²(60°))
  • ± 1/√(1 + cot²(60°))
  • ± √(sec²(60°) - 1)/sec 60°
  • 1/cosec 60°

Note: The final value of sin 60° will be positive because it is located in the first quadrant.

Trigonometric Identities for Sin 60°

We can also use trigonometric identities to express sin 60° in different forms:

  • sin(180° - 60°) = sin 120°
  • -sin(180° + 60°) = -sin 240°
  • cos(90° - 60°) = cos 30°
  • -cos(90° + 60°) = -cos 150°

Sin 60 Degrees Using the Unit Circle

To find the value of sin 60 degrees using the unit circle:

  1. Rotate a radius (r) counterclockwise to form a 60° angle with the positive x-axis.
  2. The intersection of this radius with the unit circle will be at the point (0.5, 0.866).
  3. The y-coordinate of this point (0.866) represents the sine of 60 degrees.

Therefore, sin 60° = 0.866 (approximately).

Related Links

Frequently Asked Questions on Value of Sin 60

The exact value of sin 60 degrees as a simplified fraction is √3/2

The value of sin 30 degrees is 1/2, while the value of sin 60 degrees is √3/2

No, sin 60 degrees is not equal to sin 120 degrees. However, sin 60 degrees = -sin(-60 degrees) = -sin 300 degrees.

The value of sin 60 degrees is √3/2 and the value of cos 60 degrees is 1/2.

To find the value of sin 60 degrees, you can construct a 30-60-90 triangle and use the ratio of the opposite side to the hypotenuse, which is √3/2