Sine Calculator — sin(x) in Degrees or Radians
sin(x)
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Angle in Degrees
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Angle in Radians
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Understanding the Sine Function
The sine function is one of the fundamental trigonometric functions. For an angle in a right triangle, sin(x) equals the ratio of the opposite side to the hypotenuse. The function produces values between -1 and 1 and has a period of 360 degrees (2 pi radians).
Common exact values include sin(0) = 0, sin(30) = 0.5, sin(45) = sqrt(2)/2, sin(60) = sqrt(3)/2, and sin(90) = 1. The sine function is odd, meaning sin(-x) = -sin(x).
Sine is used extensively in physics for modeling waves, oscillations, and circular motion, in engineering for signal processing and AC circuits, and in navigation for calculating distances and bearings.
Frequently Asked Questions
What is the range of the sine function?
The sine function outputs values between -1 and 1, inclusive. sin(x) = 1 when x = 90 degrees and sin(x) = -1 when x = 270 degrees.
What is the difference between degrees and radians?
Degrees divide a full circle into 360 parts. Radians measure angles by the arc length on a unit circle. A full circle is 2 pi radians. To convert: radians = degrees times pi/180.
What is sin(45 degrees)?
sin(45 degrees) = sqrt(2)/2, which is approximately 0.7071. This is one of the standard exact values that appears frequently in geometry, physics, and engineering. It represents the ratio in an isosceles right triangle where both legs are equal length. Other key exact values: sin(0) = 0, sin(30) = 0.5, sin(60) = sqrt(3)/2 (approximately 0.8660), and sin(90) = 1.
What is the relationship between sine and cosine?
Sine and cosine are complementary functions: sin(x) = cos(90 - x) and cos(x) = sin(90 - x). They are also related by the Pythagorean identity: sin squared(x) + cos squared(x) = 1 for all values of x. On the unit circle, sin gives the y-coordinate and cos gives the x-coordinate of a point at angle x from the positive x-axis. The cosine function is identical to the sine function shifted 90 degrees to the left.
Where is the sine function used in real life?
The sine function models any periodic wave-like phenomenon. In physics, it describes sound waves, light waves, alternating current (AC) electricity, and pendulum motion. In engineering, it is essential for signal processing, vibration analysis, and structural load calculations. In navigation, sine appears in the law of sines for calculating distances and bearings. In music, sine waves represent pure tones, and complex sounds are decomposed into sine components using Fourier analysis.