What Are The Output Values Of Tangent? (5 Key Values To Know)

In trigonometry, you will sometimes see common input angles (such as 45 or 90 degrees) for the tangent function.  It is helpful to become familiar with the corresponding outputs for these angles so you can do calculations faster.

So, what are the output values of tangent?  Five output values of tangent you should know are tan(0) = 0, tan(π/6) = √3/3, tan(π/4) = 1, tan(π/3) = √3, and tan(π/2) is undefined. 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 negative of the above outputs.

Of course, when the input angle for tangent is in quadrants 2 or 4 (angles between 90^o and 180^o, or π/2 to 3π/2 radians, and 270^o and 360^o, or 3π/2 to 2π radians), we get negative values for tangent.  When the input angle for tangent is in quadrants 1 or 3 (angles between 0^o and 90^o or 180^o and 270^o), we get positive values for tangent.

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

Let’s get started.

What Are The Output Values Of Tangent?

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

Angle [A]	Output Value Of Tangent [tan(A)]
0	0
30	√3/3
45	1
60	√3
90	undefined

There are a couple of different ways to remember or derive these values.

One way is to remember that the tangent of an angle A is the same as the sine of the angle A divided by the cosine of the angle A, or:

• tan(A) = sin(A)/cos(A)

This is because of SOH-CAH-TOA: tangent is opposite over adjacent, and sine is opposite over hypotenuse, while cosine is adjacent over hypotenuse.

So, for example:

• tan(60) = sin(60) / cos(60) = (√3/2) / (1/2) = √3

As another way to derive values, remember that tan(90 – A) is equal to 1/tan(A). This is because of the complementary sine and cosine identities, which state that:

• sin(90 – A) = cos(A)
• cos(90 – A) = sin(A)

Using both of the above identities, we can prove that:

• tan(90 – A)
• =sin(90 – A) / cos(90 – A)
• =cos(A) / sin(A)
• = 1/tan(A)

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 tangent are negative in the 2nd quadrant.

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

Angle [A]	Reference Angle [A’]	Output Value Of Tangent [tan(A)]
180	0	0
150	30	-√3/3
135	45	-1
120	60	-√3
90	90	undefined

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

Also, the output values of tangent are positive in the 3rd quadrant (since both sine and cosine are negative in the 3rd quadrant, so their ratio is positive).

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

Angle [A]	Reference Angle [A’]	Output Value Of Tangent [tan(A)]
180	0	0
210	30	√3/3
225	45	1
240	60	√3
270	90	undefined

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

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

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

Angle [A]	Reference Angle [A’]	Output Value Of Tangent [tan(A)]
360	0	0
330	30	-√3/3
315	45	-1
300	60	-√3
270	90	undefined

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 Tangent

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

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

So:

• tan(A)
• =tan(60^o)
• =√3

Example 2: Tangent 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 tangent 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 tangent has negative output in the 2^nd quadrant.  Now, we need to look up the value in the first table above.

So:

• tan(A)
• =tan(150^o)
• =-tan(30^o)  [using our reference angle, A’ = 30^o, and tangent is negative in the 2^nd quadrant]
• =-√3/3  [tan(30^o) = √3/3, from the table above]

Example 3: Tangent 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 tangent 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 tangent has positive output in the 3rd quadrant.  Now, we need to look up the value in the first table above.

So:

• tan(A)
• =tan(225^o)
• =tan(45^o)  [using our reference angle, A’ = 45^o, and tangent is positive in the 3^rd quadrant]
• =1  [tan(45^o) = 1, from the table above]

Example 4: Tangent 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 tangent 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 tangent has negative output in the 4th quadrant.  Now, we need to look up the value in the first table above.

So:

• tan(A)
• =tan(330^o)
• =-tan(30^o)  [using our reference angle, A’ = 30^o, and tangent is negative in the 4^th quadrant]
• =-√3/3  [tan(30^o) = √3/3, from the table above]

Example 5: Tangent 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 tangent of the angle A = -150^o, we can find the tangent 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 tangent has positive output in the 3rd quadrant.  Now, we need to look up the value in the first table above.

So:

• tan(A)
• =tan(B)
• =tan(210^o)
• =tan(30^o)  [using our reference angle, B’ = 30^o, and tangent is positive in the 3^rd quadrant]
• =√3/3  [tan(30^o) = √3/3, from the table above]

Example 6: Tangent 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 tangent of the angle A = 630^o, we can find the tangent 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 tangent has positive output in the 3rd quadrant.  Now, we need to look up the value in the first table above.

So:

• tan(A)
• =tan(B)
• =tan(270^o)
• =tan(90^o)  [using our reference angle, B’ = 90^o, and tangent is positive in the 3^rd quadrant]
• =undefined  [tan(90^o) is undefined, from the table above]

Conclusion

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

You can learn more about uses of trig functions here.

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