In trigonometry, you will sometimes see common input angles (such as 45 or 90 degrees) for the cosine function. It helps to know the corresponding outputs for these angles to help you do calculations faster.

So, what are the output values of cosine? **Five output values of cosine you should know are cos(0) = 1, cos(π/6) = √3/2, cos(π/4) = √2/2, cos(π/3) = 1/2, and cos(π/2) = 0 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 same outputs.**

Of course, when the input angle for cosine is in quadrants 2 or 3 (angles between 90^{o} and 270^{o}, or π/2 to 3π/2 radians), we get negative values for cosine. When the input angle for cosine is in quadrants 1 or 4 (angles between 0^{o} and 90^{o} or 270^{o} and 360^{o}), we get positive values for cosine.

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

Let’s get started.

## What Are The Output Values Of Cosine?

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

Angle [A] | Output Value Of Cosine [cos(A)] |
---|---|

0 | 1 |

30 | √3/2 |

45 | √2/2 |

60 | 1/2 |

90 | 0 |

angles in the 1st quadrant

and corresponding output

values for the cosine function.

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

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

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 cosine are negative in the 2nd quadrant.

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

Angle [A] | Reference Angle [A’] | Output Value Of Cosine [cos(A)] |
---|---|---|

180 | 0 | -1 |

150 | 30 | -√3/2 |

135 | 45 | -√2/2 |

120 | 60 | -1/2 |

90 | 90 | 0 |

2nd quadrant and corresponding reference

angles and output values for the cosine 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 cosine are negative in the 3rd quadrant.

So, in the 3rd quadrant, we would get the following reference angles and output values for cosine:

Angle [A] | Reference Angle [A’] | Output Value Of Cosine [cos(A)] |
---|---|---|

180 | 0 | -1 |

210 | 30 | -√3/2 |

225 | 45 | -√2/2 |

240 | 60 | -1/2 |

270 | 90 | 0 |

3rd quadrant and corresponding reference

angles and output values for the cosine 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 cosine are positive in the 4th quadrant.

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

Angle [A] | Reference Angle [A’] | Output Value Of Cosine [cos(A)] |
---|---|---|

360 | 0 | 1 |

330 | 30 | √3/2 |

315 | 45 | √2/2 |

300 | 60 | 1/2 |

270 | 90 | 0 |

4th quadrant and corresponding reference

angles and output values for the cosine 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 Cosine

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

#### Example 1: Cosine 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 cosine of the angle A, all we need to do is look up the value in the first table above.

So:

**cos(A)****=cos(60**^{o})**=1/2**

#### Example 2: Cosine 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 cosine 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 cosine has negative output in the 2^{nd} quadrant. Now, we need to look up the value in the first table above.

So:

**cos(A)****=cos(150**^{o})**=-cos(30**[using our reference angle, A’ = 30^{o})^{o}, and cosine is negative in the 2^{nd}quadrant]**=-√3/2**[cos(30^{o}) = √3/2, from the table above]

#### Example 3: Cosine 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 cosine 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 cosine has negative output in the 3rd quadrant. Now, we need to look up the value in the first table above.

So:

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

#### Example 4: Cosine 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 cosine 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 cosine has positive output in the 4th quadrant. Now, we need to look up the value in the first table above.

So:

**cos(A)****=cos(330**^{o})**=cos(30**[using our reference angle, A’ = 30^{o})^{o}, and cosine is positive in the 4^{th}quadrant]**=√3/2**[cos(30^{o}) = √3/2, from the table above]

#### Example 5: Cosine 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 cosine of the angle A = -150^{o}, we can find the cosine 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 cosine has negative output in the 3rd quadrant. Now, we need to look up the value in the first table above.

So:

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

#### Example 6: Cosine 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 cosine of the angle A = 630^{o}, we can find the cosine 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 cosine has negative output in the 3rd quadrant. Now, we need to look up the value in the first table above.

So:

**cos(A)****=cos(B)****=cos(270**^{o})**=-cos(90**[using our reference angle, B’ = 90^{o})^{o}, and cosine is negative in the 3^{rd}quadrant]**=0**[cos(90^{o}) = 0, from the table above]

## Conclusion

Now you know more about the output values of cosine 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 about the output values of sine here.

You can learn more about uses of trig functions here.

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