Sin Theta Calculator

Created by Luciano Miño
Reviewed by Luis Hoyos
Last updated: Jan 24, 2023

The sin theta calculator can aid you in your math problems and obtain the sine value of any angle.

We've paired this calculator with a few paragraphs covering:

  • What is the definition of sin (sine definition);
  • What sin of 0 is;
  • Properties of sine; and
  • More about this particular trigonometric function 📐

Keep reading to learn more!

What is the definition of sin/sine?

Before we start with the sine function definition, we need to introduce the unit circle. This circle is centered at the origin, and its radius equals one.

If we draw a line from the origin to any point on this unit circle, an angle theta θ\theta will be formed between this radius and the horizontal axis.

The sine, or sin, is the y-axis coordinate of this radius as the angle changes describing any point on the unit circle.

Y-axis projection of the radius at a given angle on a unitary circle.

Like cosine, sine is a periodic function with a period of . This means that for any argument θ\theta: sin(θ+2kπ)=sin(θ)\sin(\theta + 2k\pi) = \sin(\theta) where kk is any integer.

💡 Test it out! Input any angle in our sin theta calculator and write down the sine result. Now try again with the same angle, but add 2*π (or 360°, if you're using degrees) to it and see if the results match.

Properties of sine

Here's a list of some of the sine properties and trigonometric identities involving the sine function:

sin(x)\sin(-x)

=sin(x)= -\sin(x)

sin2(x)\sin^2(x)

=1cos2(x)= 1 - \cos^2(x)

sin(2x)\sin(2x)

=2sin(x)cos(x)= 2\cdot \sin(x) \cdot \cos(x)

sin(x2)\sin(\frac{x}{2})

=±1cos(x)2= \pm \sqrt{\frac{1 -\cos(x)}{2}}

sin(x+y)\sin(x+y)

=sin(x)cos(y)+cos(x)sin(y)= \sin(x)\cos(y) + \cos(x)\sin(y)

sin(xy)\sin(x-y)

=sin(x)cos(y)cos(x)sin(y)= \sin(x)\cos(y) - \cos(x)\sin(y)

ddxsin(x)\frac{d}{dx}\sin(x)

=cos(x)= \cos(x) (*)

(*) Only when using radians.

🙋 You can input the sine argument in either degrees, radians, or π radians with our sin theta calculator!

Other sine calculators

Feel free to check our other tools related to the sin theta calculator:

FAQ

What is the sin of 2 theta?

The sine of 2 theta () equals 2 sin(θ)cos(θ). According to one of the sine properties: sin(2x) = 2sin(x)cos(x). So, if you know the values for the sine and cosine of theta, you can easily find the sine of 2 theta.

How do I find the sin of theta/2?

To find the sin of theta/2:

  1. Write down the sine half-angle equation: sin(θ/2) = ±√[(1-cos(θ))/2].
  2. Replace theta θ within the equation and solve the square root.
  3. To choose the sign, follow this rule:
    • The result is positive (+) if the half angle lies in the I or the II quadrant; or
    • Negative (-) if it lies on the III or IV quadrant.

What is the sin of 0?

0. The value for the sine function at 0 is 0. The y-axis component of the radius in the unit circle is 0 when the radius lies on the horizontal axis (which corresponds to angle zero).

Luciano Miño
Angle
deg
Plot of sin(x) in <-2π, 2π> range
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