What Is The Value Of Tan 135 Degrees?
The value of tan 135 degrees is $-1$, or $-1.0000$ to four decimal places. Written with the angle in both forms, $\tan 135^\circ = \tan\frac{3\pi}{4} = -1$.
Two facts fix this value:
The angle lives in the second quadrant. $135^\circ$ is between $90^\circ$ and $180^\circ$, and in that quadrant tangent is negative.
Its reference angle is $45^\circ$. Since $\tan 45^\circ = 1$, the size of the answer is $1$, and the quadrant supplies the minus sign.
Put together, $\tan 135^\circ = -1$. The rest of this article shows four ways to reach that number and why each one agrees.
How Do You Find Tan 135 Degrees Using The Reference Angle?
The reference angle is the acute angle between the terminal arm and the x-axis. For any angle in the second quadrant, you find it by subtracting from $180^\circ$.
$$\text{Reference angle} = 180^\circ - 135^\circ = 45^\circ$$
Next, decide the sign from the quadrant. A quick way to remember it is the CAST rule (or ASTC): in the second quadrant, only Sine is positive, so cosine and tangent are both negative there.
$$\tan 135^\circ = -\tan(180^\circ - 135^\circ) = -\tan 45^\circ$$
$$\tan 135^\circ = -(1) = -1$$
That is the whole method in three lines: shrink the angle to its reference, look up the acute value, then attach the quadrant sign. It works for every special angle in the second quadrant, and you can see the family in the trigonometric ratios of specific angles.
Where Does 135° Sit On The Unit Circle?
On the unit circle, an angle is measured anticlockwise from the positive x-axis, and the point where the terminal arm meets the circle has coordinates $(\cos\theta, \sin\theta)$. For $135^\circ$, that point is:
$$\left(\cos 135^\circ,\ \sin 135^\circ\right) = \left(-\frac{\sqrt{2}}{2},\ \frac{\sqrt{2}}{2}\right) \approx (-0.7071,\ 0.7071)$$
Tangent is the ratio of the y-coordinate to the x-coordinate:
$$\tan 135^\circ = \frac{\sin 135^\circ}{\cos 135^\circ} = \frac{\tfrac{\sqrt{2}}{2}}{-\tfrac{\sqrt{2}}{2}} = -1$$
The point sits in the top-left of the circle, so its x is negative and its y is positive. A positive divided by a negative is negative, which is exactly why the tangent comes out as $-1$.
Can You Read Tan 135 Degrees From A Right Triangle?
Yes, and doing both readings is what makes the value stick. The reference triangle for $135^\circ$ is a 45-45-90 triangle, the same shape you meet at $45^\circ$. In a 45-45-90 triangle the two legs are equal, so the ratio opposite over adjacent is exactly $1$:
$$\tan 45^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{1} = 1$$
The triangle only ever gives the positive size, $1$. What the triangle cannot tell you on its own is the direction. That is the job of the quadrant: at $135^\circ$ the horizontal side points in the negative-x direction, so the signed ratio flips to $-1$.
This is the double anchor worth carrying with you. The right triangle sets the magnitude ($1$), and the unit circle sets the sign ($-$). For a refresher on how the three core ratios are defined from a triangle, see sin cos tan and the tangent function.
How Do You Derive Tan 135 Degrees With The Difference Formula?
For an exact confirmation that uses no picture, write $135^\circ$ as $180^\circ - 45^\circ$ and apply the tangent difference formula:
$$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$
Substitute $A = 180^\circ$ and $B = 45^\circ$, using $\tan 180^\circ = 0$ and $\tan 45^\circ = 1$:
$$\tan(180^\circ - 45^\circ) = \frac{\tan 180^\circ - \tan 45^\circ}{1 + \tan 180^\circ \tan 45^\circ}$$
$$= \frac{0 - 1}{1 + (0)(1)} = \frac{-1}{1} = -1$$
The algebra lands on the same answer as the unit circle and the reference angle. For the general identities behind this step, see the sum and difference identities and the worked tan a minus b formula.
What Are The Values Around Tan 135 Degrees?
Comparing $135^\circ$ with its neighbours shows how the tangent changes sign and size across the second quadrant.
Table: Sine, cosine, and tangent for angles around 135°, in degrees and radians.
Angle | Radians | $\sin$ | $\cos$ | $\tan$ |
|---|---|---|---|---|
$45^\circ$ | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ |
$120^\circ$ | $\frac{2\pi}{3}$ | $\frac{\sqrt{3}}{2}$ | $-\frac{1}{2}$ | $-\sqrt{3}$ |
$135^\circ$ | $\frac{3\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-1$ |
$150^\circ$ | $\frac{5\pi}{6}$ | $\frac{1}{2}$ | $-\frac{\sqrt{3}}{2}$ | $-\frac{1}{\sqrt{3}}$ |
$180^\circ$ | $\pi$ | $0$ | $-1$ | $0$ |
Reading down the tangent column, every value from $120^\circ$ to $150^\circ$ is negative, and the size shrinks from $\sqrt{3}$ toward $0$ as the angle approaches $180^\circ$. The full set lives in the trigonometric table. The radian form of $135^\circ$ has its own reference page at tan 3pi/4.
Why Is Tan 135 Degrees Negative?
The sign is not a rule to memorise. It falls straight out of where the angle points.
The terminal arm lands in the second quadrant. There the x-coordinate is negative and the y-coordinate is positive.
Tangent is y divided by x. A positive number over a negative number is negative, so tangent must be negative here.
The magnitude is set by the reference angle. The $45^\circ$ reference gives a size of $1$, and the quadrant then stamps the minus sign onto it.
The same reasoning explains why cosine is negative at $135^\circ$ while sine stays positive: cosine reads the negative x-coordinate, sine reads the positive y-coordinate. You can see the paired value at cos 135° and the radian versions at sin 3pi/4 and cos 3pi/4.
Who Discovered The Angle Values Behind Tan 135 Degrees?
Long before calculators, astronomers built trigonometry to measure the sky, and the tables they wrote are the ancestors of every special-angle value we use today.
Two later mathematicians carried the work forward:
Claudius Ptolemy (c. 100 – c. 170 CE, Roman Egypt) collected and extended the chord tables in the Almagest, the astronomy reference that shaped the field for over a thousand years.
Aryabhata (476 – 550 CE, India) compiled one of the earliest sine tables and used the term jya, which travelled through Arabic into the Latin word sinus, the root of our word "sine."
Where Is Tan 135 Degrees Used In The Real World?
A $135^\circ$ direction and a slope of $-1$ turn up wherever something rises while heading the opposite way.
Navigation and bearings: a heading of $135^\circ$ points to the south-east, and pilots and sailors read these angles directly off a compass to hold a course.
Computer graphics: rotating a sprite or camera by $135^\circ$ uses the sine and cosine of the angle, and the $-1$ tangent describes a diagonal that runs down-right to up-left across the screen.
Physics and projectiles: the launch and landing angles of a symmetric arc are supplementary, so a $45^\circ$ launch pairs with a $135^\circ$ line of descent.
Engineering and design: a line with slope $-1$ (a $135^\circ$ incline from the horizontal) is the standard 45-degree back-slope used for chamfers, roof valleys, and ramp cutaways.
One angle, read as a bearing, a rotation, a trajectory, or a slope, keeps its tangent of $-1$ across every one of these fields.
What Are The Most Common Mistakes With Tan 135 Degrees?
These four slips account for most wrong answers on this angle, and each has a clean fix.
Dropping the negative sign.
Where it slips in:
A student finds the reference value $\tan 45^\circ = 1$ and writes $\tan 135^\circ = 1$, forgetting the quadrant.
Don't do this:
Do not report the reference-angle value as the final answer. The reference angle only gives the size.
The correct way:
Check the quadrant before writing the answer. $135^\circ$ is in the second quadrant, where tangent is negative, so $\tan 135^\circ = -1$.
Leaving the calculator in the wrong mode.
Where it slips in:
A student types 135 while the calculator is set to radians and reads off a strange decimal instead of $-1$.
Don't do this:
Do not trust the display until the angle mode matches the angle. In radian mode, "135" means 135 radians, not $135^\circ$.
The correct way:
Set the calculator to degree mode for $135^\circ$, or enter $\frac{3\pi}{4}$ in radian mode. Both return $-1$. If you are unsure what a radian is, start with what is a radian.
Finding the reference angle from the wrong side.
Where it slips in:
A student computes $135^\circ - 90^\circ = 45^\circ$ and happens to get the right number, then uses $\theta - 90^\circ$ on a different Q2 angle and gets it wrong.
Don't do this:
Do not subtract $90^\circ$ for second-quadrant angles. That shortcut is not the reference-angle rule.
The correct way:
For a second-quadrant angle, always use $180^\circ - \theta$. Here $180^\circ - 135^\circ = 45^\circ$, which is the true reference angle.
Mixing up which ratio changes sign.
Where it slips in:
A student assumes sine, cosine, and tangent all share one sign at $135^\circ$.
Don't do this:
Do not give every ratio the same sign. At $135^\circ$, sine is positive while cosine and tangent are negative.
The correct way:
Read each ratio from the coordinates. With the point at $(-\tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{2}}{2})$, sine (the y-value) is positive and cosine (the x-value) is negative, so their ratio, the tangent, is negative.
Practice Problems On Tan 135 Degrees
Work each one, then check the answer beside it.
State the reference angle of $135^\circ$.
(Answer: $180^\circ - 135^\circ = 45^\circ$.)Write $135^\circ$ in radians.
(Answer: $\frac{3\pi}{4}$.)Evaluate $\dfrac{\sin 135^\circ}{\cos 135^\circ}$.
(Answer: $\dfrac{\sqrt{2}/2}{-\sqrt{2}/2} = -1$.)Find $\tan 135^\circ + \tan 45^\circ$.
(Answer: $-1 + 1 = 0$.)Is $\tan 135^\circ$ greater than or less than $\tan 120^\circ$?
(Answer: $-1 > -\sqrt{3}$, so $\tan 135^\circ$ is greater.)Use $\tan(180^\circ - \theta) = -\tan\theta$ to confirm $\tan 135^\circ$.
(Answer: $-\tan 45^\circ = -1$.)
Where Should You Go Next After Tan 135 Degrees?
Once the value of $\tan 135^\circ$ is clear, a few natural doors open from here.
Trigonometric ratios. Ground the six ratios so any special angle becomes a lookup plus a sign.
Unit circle with tangent. See how tangent behaves all the way around the circle, including where it shoots to infinity.
Tan 45 degrees. Master the reference value that every second-quadrant tangent of this family depends on.
If your child is building these foundations, a live Bhanzu trainer teaches special-angle values starting from the reference angle and the unit circle in the Bhanzu trigonometry program.
Was this article helpful?
Your feedback helps us write better content