Tan 3pi/4 : Exact Value, −1, and How to Find It

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.

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

1. Evaluate $\tan\left(\dfrac{3\pi}{4}\right) + \tan\left(\dfrac{\pi}{4}\right)$.

2. A line makes an angle of $\dfrac{3\pi}{4}$ with the positive $x$-axis. State its slope and say why it is negative.

3. Show that $\tan\left(\dfrac{3\pi}{4}\right) \times \tan\left(\dfrac{\pi}{4}\right) = -1$.

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Read More

• Sin, cos, and tan — the three core trigonometric ratios

• Trigonometric ratios of specific angles

• The full trigonometric table of standard values

• What a radian is and how radian measure works