What Does Tan 3pi/4 Mean?
A radian is the angle that wraps an arc equal in length to the radius; $\dfrac{3\pi}{4}$ radians is three-quarters of the way from $0$ to $\pi$, landing at $135^\circ$. The four quadrants divide the plane, and $\dfrac{3\pi}{4}$ sits in the second quadrant, upper-left, where $x$ is negative and $y$ is positive.
The tangent of an angle is the slope of the radius to that point, or $\dfrac{\sin\theta}{\cos\theta}$. The reference angle is the acute angle between the radius and the $x$-axis; for $\dfrac{3\pi}{4}$ it is $\pi - \dfrac{3\pi}{4} = \dfrac{\pi}{4}$. The reference angle sets the size of the tangent; the quadrant sets the sign.
Where Does Tan 3pi/4 Show Up?
An angle of $\dfrac{3\pi}{4}$ points up and to the left along the diagonal, and its slope is $\tan\left(\dfrac{3\pi}{4}\right) = -1$ - a line that falls exactly one unit for every unit it moves right, the mirror image of the $45^\circ$ "even diagonal." Any line with gradient $-1$ makes this angle with the positive $x$-axis.
The value appears when a rotation of $135^\circ$ shows up: the perpendicular diagonals of a square meet symmetric $45^\circ$ and $135^\circ$ lines, and reflected light or a bouncing path often turns through such angles. The unit circle keeps the sign consistent across all four quadrants.
Standard-Angle Reference Table
The angle $\dfrac{3\pi}{4}$ radians is $135^\circ$ - a second-quadrant angle. This table lists the common radian angles around it so you can see where the tangent changes sign.
Angle (radians) | Angle (degrees) | $\tan\theta$ (exact) | $\tan\theta$ (decimal) |
|---|---|---|---|
$\dfrac{\pi}{6}$ | $30^\circ$ | $\dfrac{1}{\sqrt{3}}$ | $0.5774$ |
$\dfrac{\pi}{4}$ | $45^\circ$ | $1$ | $1.0000$ |
$\dfrac{\pi}{3}$ | $60^\circ$ | $\sqrt{3}$ | $1.7321$ |
$\dfrac{\pi}{2}$ | $90^\circ$ | undefined | — |
$\dfrac{2\pi}{3}$ | $120^\circ$ | $-\sqrt{3}$ | $-1.7321$ |
$\dfrac{3\pi}{4}$ | $135^\circ$ | $-1$ | $-1.0000$ |
Notice the sign flip after $\dfrac{\pi}{2}$: once the angle crosses into the second quadrant, tangent turns negative. Tan 3π/4 carries the same magnitude as tan π/4 but the opposite sign, because $\dfrac{\pi}{4}$ is its reference angle.
How Do You Find The Exact Value Of Tan 3pi/4?
There are two clean routes, and both give $-1$.
Method 1: Reference angle plus quadrant sign.
First find the reference angle: $\pi - \dfrac{3\pi}{4} = \dfrac{\pi}{4}$. The tangent of the reference angle is $\tan\left(\dfrac{\pi}{4}\right) = 1$.
Now apply the sign. In the second quadrant sine is positive and cosine is negative, so their ratio — the tangent — is negative:
$$\tan\left(\frac{3\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right) = -1$$
Method 2: The unit-circle coordinates.
At $\dfrac{3\pi}{4}$ the radius meets the unit circle at $\left(-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)$. Tangent is $\dfrac{y}{x}$:
$$\tan\left(\frac{3\pi}{4}\right) = \frac{\sin(3\pi/4)}{\cos(3\pi/4)} = \frac{\sqrt{2}/2}{-\sqrt{2}/2} = -1$$
Both routes agree, and a calculator in radian mode confirms it: $\tan(3\pi/4) = -1$ exactly.
Examples Of Tan 3pi/4
Example 1
Evaluate $8\tan\left(\dfrac{3\pi}{4}\right)$.
$$8\tan\left(\frac{3\pi}{4}\right) = 8 \times (-1) = -8$$
Example 2
Find $\tan\left(\dfrac{3\pi}{4}\right)$ using its reference angle.
Wrong attempt. A student finds the reference angle $\dfrac{\pi}{4}$ and stops at $\tan\left(\dfrac{\pi}{4}\right) = 1$, reporting $+1$.
That misses the quadrant. The reference angle only fixes the magnitude; $\dfrac{3\pi}{4}$ sits in the second quadrant, where tangent is negative, so a positive answer contradicts the coordinates $\left(-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)$.
Correct. Keep the magnitude $1$ from the reference angle, then attach the second-quadrant sign: $\tan\left(\dfrac{3\pi}{4}\right) = -1$.
Example 3
A line through the origin makes an angle of $\dfrac{3\pi}{4}$ with the positive $x$-axis. What is its slope?
$$\text{slope} = \tan\left(\frac{3\pi}{4}\right) = -1$$
The slope of $-1$ means the line falls one unit for every unit to the right — the perpendicular mirror of a $45^\circ$ line.
Example 4
Verify that $\tan\left(\dfrac{3\pi}{4}\right) = \dfrac{\sin(3\pi/4)}{\cos(3\pi/4)}$.
$$\frac{\sin(3\pi/4)}{\cos(3\pi/4)} = \frac{\sqrt{2}/2}{-\sqrt{2}/2} = -1$$
The quotient identity holds, and the equal-but-opposite coordinates make the ratio exactly $-1$.
Example 5
Convert $\dfrac{3\pi}{4}$ to degrees and confirm the value.
Since $\pi$ radians $= 180^\circ$, $\dfrac{3\pi}{4} = \dfrac{3 \times 180^\circ}{4} = 135^\circ$, and $\tan 135^\circ = -1$. The radian and degree forms name the same angle and the same value.
Where Students Trip Up On Tan 3pi/4
Mistake 1: Dropping the negative sign
Where it slips in: Computing the reference-angle tangent and forgetting to apply the quadrant sign.
Don't do this: Writing $\tan\left(\dfrac{3\pi}{4}\right) = 1$.
The correct way: The reference angle gives the size $1$; the second quadrant makes it negative, so the answer is $-1$. The habit that prevents this is naming the quadrant before writing the sign, rather than after.
Mistake 2: Using the wrong reference angle
Where it slips in: Subtracting from the wrong axis, so the reference angle comes out as $\dfrac{3\pi}{4}$ or some other value.
Don't do this: Treating $\dfrac{3\pi}{4}$ as its own reference angle.
The correct way: In the second quadrant the reference angle is $\pi - \theta$, so $\pi - \dfrac{3\pi}{4} = \dfrac{\pi}{4}$. That is why tan 3π/4 shares its magnitude with $\tan\left(\dfrac{\pi}{4}\right)$, the value covered in tan 45 degrees.
Mistake 3: Confusing the magnitude with sin 3π/4 or cos 3π/4
Where it slips in: Reaching for the $\dfrac{\sqrt{2}}{2}$ that sine and cosine carry at this angle.
Don't do this: Writing $\tan\left(\dfrac{3\pi}{4}\right) = \dfrac{\sqrt{2}}{2}$.
The correct way: $\sin\left(\dfrac{3\pi}{4}\right) = \dfrac{\sqrt{2}}{2}$ and $\cos\left(\dfrac{3\pi}{4}\right) = -\dfrac{\sqrt{2}}{2}$, but tangent is their ratio, and equal-magnitude opposite-sign values divide to $-1$, not $\dfrac{\sqrt{2}}{2}$.
Key Takeaways
Tan 3pi/4 equals exactly $-1$ — the same magnitude as $\tan\left(\dfrac{\pi}{4}\right)$, with the negative sign of the second quadrant.
The reference angle is $\pi - \dfrac{3\pi}{4} = \dfrac{\pi}{4}$, which sets the magnitude $1$; the second quadrant sets the sign.
In degrees, $\tan\left(\dfrac{3\pi}{4}\right) = \tan 135^\circ$, and on the unit circle the point is $\left(-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)$.
The most common slip is dropping the negative sign and reporting $+1$ — always name the quadrant before writing the sign.
To take radian-measure angles further with a teacher, explore Bhanzu's trigonometry tutor, our high school math tutor sessions, or structured math classes online.
Practice These Before Moving On
Evaluate $\tan\left(\dfrac{3\pi}{4}\right) + \tan\left(\dfrac{\pi}{4}\right)$.
A line makes an angle of $\dfrac{3\pi}{4}$ with the positive $x$-axis. State its slope and say why it is negative.
Show that $\tan\left(\dfrac{3\pi}{4}\right) \times \tan\left(\dfrac{\pi}{4}\right) = -1$.
Want a live Bhanzu trainer to walk through more tan 3pi/4 problems? Book a free demo class — online, worldwide.
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