In trigonometry, we often use common input angles (such as 30 or 60 degrees) for the sine function. It helps to know the corresponding outputs for these common angles to increase your calculation speed.

So, what are the output values of sine? **Five output values of sine you should know are sin(0) = 0, sin(π/6) = 1/2, sin(π/4) = √2/2, sin(π/3) = √3/2, and sin(π/2) = 1. These output values correspond to the common degree measures 0 ^{o}, 30^{o}, 45^{o}, 60^{o}, and 90^{o}. Using the negative of these angles as input gives us the opposite (negative) outputs.**

Of course, when the input angle for sine is in quadrants 1 or 2 (angles between 0^{o} and 180^{o}, or 0 to π radians), we get positive values for sine. When the input angle for sine is in quadrants 3 or 4 (angles between 180^{o} and 360^{o}, or π to 2π radians), we get negative values for sine.

In this article, we’ll talk about output values for sine and how to remember the 5 common ones mentioned above. We’ll also review the signs of the outputs and when they are positive, negative, or zero.

Let’s get started.

## What Are The Output Values Of Sine?

The output values of sine are given in the table below for 5 common values:

Angle [A] | Output Value Of Sine [sin(A)] |
---|---|

0 | 0 |

30 | 1/2 |

45 | √2/2 |

60 | √3/2 |

90 | 1 |

angles in the 1st quadrant

and corresponding output

values for the sine function.

As an easy way to remember the values, note that each output value has the form √n/2, where n goes from 0 to 4 as we move from the degree measures 0 to 90^{o} (0 to π/2 radians):

**For 0**^{o}: use n = 0, so the output is sin(0^{o}) = √0/2 = 0/2 = 0.**For 30**^{o}: use n = 1, so the output is sin(30^{o}) = √1/2 = 1/2.**For 45**^{o}: use n = 2, so the output is sin(45^{o}) = √2/2.**For 60**^{o}: use n = 3, so the output is sin(60^{o}) = √3/2.**For 90**^{o}: use n = 4, so the output is sin(90^{o}) = √4/2 = 2/2 = 1.

Remember that we measure reference angles from the x-axis to get an acute angle (less than 90 degrees). This means that in the 2nd quadrant, we subtract the angle from 180 degrees (π radians) to get our reference angle.

Also, keep in mind that the output values of sine are positive in the 2^{nd} quadrant.

So, in the 2^{nd} quadrant, we would get the following reference angles and output values for sine:

Angle [A] | Reference Angle [A’] | Output Value Of Sine [sin(A)] |
---|---|---|

180 | 0 | 0 |

150 | 30 | 1/2 |

135 | 45 | √2/2 |

120 | 60 | √3/2 |

90 | 90 | 1 |

2nd quadrant and corresponding reference

angles and output values for the sine function.

In the 3^{rd} quadrant, we subtract 180 degrees (π radians) from the angle to get our reference angle.

Also, the output values of sine are negative in the 3rd quadrant.

In the 3rd quadrant, we would get the following reference angles and output values for sine:

Angle [A] | Reference Angle [A’] | Output Value Of Sine [sin(A)] |
---|---|---|

180 | 0 | 0 |

210 | 30 | -1/2 |

225 | 45 | -√2/2 |

240 | 60 | -√3/2 |

270 | 90 | -1 |

3rd quadrant and corresponding reference

angles and output values for the sine function.

In the 4^{th} quadrant, we subtract the angle from 360 degrees (2π radians) to get our reference angle.

Also, the output values of sine are negative in the 4th quadrant.

In the 4th quadrant, we would get the following reference angles and output values for sine:

Angle [A] | Reference Angle [A’] | Output Value Of Sine [sin(A)] |
---|---|---|

360 | 0 | 0 |

330 | 30 | -1/2 |

315 | 45 | -√2/2 |

300 | 60 | -√3/2 |

270 | 90 | -1 |

4th quadrant and corresponding reference

angles and output values for the sine function.

Finally, remember that for a negative angle, we add 360 degrees (2π radians) until we get to an angle between 0^{o} and 360^{o}.

For an angle greater than 360^{o}, we do the opposite and subtract 360 degrees (2π radians) until we get to an angle between 0^{o} and 360^{o}.

### Examples Of Output Values Of Sine

Let’s take a look at some examples to show how to find the output values of sine for various input angles.

#### Example 1: Sine In The First Quadrant

Let’s say that we have the angle A = 60^{o}, which is in the first quadrant (it is between 0^{o} and 90^{o}).

To find the sine of the angle A, all we need to do is look up the value in the first table above.

So:

**sin(A)****=sin(60**^{o})**=√3/2**

#### Example 2: Sine In The Second Quadrant

Let’s say that we have the angle A = 150^{o}, which is in the second quadrant (it is between 90^{o} and 180^{o}).

To find the sine of the angle A, we will first need to find the reference angle A’. In the second quadrant, we calculate

- A’ = 180
^{o}– A - A’ = 180
^{o}– 150^{o} - A’ = 30
^{o}

Also, we know that sine has positive output in the 2^{nd} quadrant. Now, we need to look up the value in the first table above.

So:

**sin(A)****=sin(150**^{o})**=sin(30**[using our reference angle, A’ = 30^{o})^{o}, and sine is positive in the 2^{nd}quadrant]**=1/2**[sin(30^{o}) = 1/2, from the table above]

#### Example 3: Sine In The Third Quadrant

Let’s say that we have the angle A = 225^{o}, which is in the third quadrant (it is between 180^{o} and 270^{o}).

To find the sine of the angle A, we will first need to find the reference angle A’. In the third quadrant, we calculate

- A’ = A – 180
^{o} - A’ = 225
^{o}– 180^{o} - A’ = 45
^{o}

Also, we know that sine has negative output in the 3rd quadrant. Now, we need to look up the value in the first table above.

So:

**sin(A)****=sin(225**^{o})**=-sin(45**[using our reference angle, A’ = 45^{o})^{o}, and sine is negative in the 3^{rd}quadrant]**=-√2/2**[sin(45^{o}) = √2/2, from the table above]

#### Example 4: Sine In The Fourth Quadrant

Let’s say that we have the angle A = 330^{o}, which is in the fourth quadrant (it is between 270^{o} and 360^{o}).

To find the sine of the angle A, we will first need to find the reference angle A’. In the fourth quadrant, we calculate

- A’ = 180
^{o}– A - A’ = 360
^{o}– 330^{o} - A’ = 30
^{o}

Also, we know that sine has negative output in the 4th quadrant. Now, we need to look up the value in the first table above.

So:

**sin(A)****=sin(330**^{o})**=-sin(30**[using our reference angle, A’ = 30^{o})^{o}, and sine is negative in the 4^{th}quadrant]**=-1/2**[sin(30^{o}) = 1/2, from the table above]

#### Example 5: Sine Of A Negative Angle

Let’s say that we have the angle A = -150^{o}, which is negative. So, we will add 360^{o} until we get an angle measure between 0^{o} and 360^{o}.

This gives us B = -150^{o} + 360^{o} = 210^{o}.

To find the sine of the angle A = -150^{o}, we can find the sine of angle B = 210^{o} (note that B = 210^{o} is in the 3^{rd} quadrant, since it is between 180^{o} and 270^{o}).

First, we need to find the reference angle B’. In the 3rd quadrant, we calculate

- B’ = B – 180
^{o} - B’ = 210
^{o}– 180^{o} - B’ = 30
^{o}

Also, we know that sine has negative output in the 3rd quadrant. Now, we need to look up the value in the first table above.

So:

**sin(A)****=sin(B)****=sin(210**^{o})**=-sin(30**[using our reference angle, B’ = 30^{o})^{o}, and sine is negative in the 3^{rd}quadrant]**=-1/2**[sin(30^{o}) = 1/2, from the table above]

#### Example 6: Sine Of An Angle Greater Than 360^{o}

Let’s say that we have the angle A = 630^{o}, which is greater than 360^{o}. So, we will subtract 360^{o} until we get an angle measure between 0^{o} and 360^{o}.

This gives us B = 630^{o} – 360^{o} = 270^{o}.

To find the sine of the angle A = 630^{o}, we can find the sine of angle B = 270^{o} (note that B = 270^{o} is on the border between the 3^{rd} and 4^{th} quadrants, since 270^{o} is the cutoff).

First, we need to find the reference angle B’. We calculate

- B’ = B – 180
^{o} - B’ = 270
^{o}– 180^{o} - B’ = 90
^{o}

Also, we know that sine has negative output in the 3rd quadrant. Now, we need to look up the value in the first table above.

So:

**sin(A)****=sin(B)****=sin(270**^{o})**=-sin(90**[using our reference angle, B’ = 90^{o})^{o}, and sine is negative in the 3^{rd}quadrant]**=-1**[sin(90^{o}) = 1, from the table above]

## Conclusion

Now you know more about the output values of sine and how they change as input angles change. You also know how to figure out the sign of the outputs, based on the quadrant of the input angle.

You can learn about the output values of cosine here.

You can learn more about uses of trig functions here.

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