Cot 3pi/4: Exact Value (-1) Explained

What Is The Value Of Cot 3pi/4?

The value of Cot 3pi/4 is $-1$. In exact form $\cot\frac{3\pi}{4} = -1$, and written as a decimal to four places it is $-1.0000$. Because $-1$ is already a whole number, the exact form and the decimal agree perfectly.

The angle appears in two equivalent forms, and Bhanzu always shows both:

• Radian form: $\frac{3\pi}{4}$, which is how the angle is written in the keyword and on the unit circle.

• Degree form: $135^\circ$, since $\frac{3\pi}{4} \times \frac{180^\circ}{\pi} = 135^\circ$.

Cotangent is one of the three reciprocal trigonometric ratios. For any angle $\theta$, it is defined as the cosine over the sine, or equally as the reciprocal of the tangent:

$$\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{1}{\tan\theta}$$

Every method below leads to the same answer, $-1$. What changes is which fact you lean on: a reference angle, a pair of coordinates on the unit circle, or the value of the tangent.

How Do You Find The Exact Value Of Cot 3pi/4?

The reference-angle method is the fastest route, and it works in two moves.

First, find the reference angle, the acute angle between the terminal side and the horizontal axis. For $\frac{3\pi}{4}$ (which lies between $\frac{\pi}{2}$ and $\pi$), the reference angle is:

$$\pi - \frac{3\pi}{4} = \frac{\pi}{4}$$

Second, fix the sign from the quadrant. The angle $\frac{3\pi}{4}$ terminates in Quadrant II. Under the ASTC rule (All, Sine, Tangent, Cosine positive by quadrant), only sine and cosecant are positive in the second quadrant. Tangent and its reciprocal cotangent are both negative there.

So the value is the first-quadrant cotangent with a minus sign attached:

$$\cot\frac{3\pi}{4} = -\cot\frac{\pi}{4} = -(1) = -1$$

The value $\cot\frac{\pi}{4} = 1$ is a special-angle fact worth memorising; see the cot pi/4 reference for its derivation from the 45-45-90 triangle.

A second route: cosine over sine. If you prefer to build cotangent from scratch, use the coordinate values directly. At $\frac{3\pi}{4}$:

$$\cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2}, \qquad \sin\frac{3\pi}{4} = \frac{\sqrt{2}}{2}$$

Divide them, and the shared $\frac{\sqrt{2}}{2}$ cancels:

$$\cot\frac{3\pi}{4} = \frac{\cos\frac{3\pi}{4}}{\sin\frac{3\pi}{4}} = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1$$

You can check either route against the cos 3pi/4 and sin 3pi/4 pages, which derive those two coordinate values on their own.

A third route: reciprocal of tangent. Since $\cot\theta = \frac{1}{\tan\theta}$ and $\tan\frac{3\pi}{4} = -1$:

$$\cot\frac{3\pi}{4} = \frac{1}{\tan\frac{3\pi}{4}} = \frac{1}{-1} = -1$$

The tan 3pi/4 page works that tangent out in full, and the reciprocal identities reference collects all three reciprocal pairs in one place.

Where Does 3pi/4 Sit On The Unit Circle?

On the unit circle, an angle is measured counterclockwise from the positive $x$-axis, and the point where the terminal side crosses the circle has coordinates $(\cos\theta, \sin\theta)$. For a full primer on radian measure, see what is a radian.

The angle $\frac{3\pi}{4}$ lands in the upper-left of the circle, at:

$$\left(-\frac{\sqrt{2}}{2}, ; \frac{\sqrt{2}}{2}\right) \approx (-0.7071, ; 0.7071)$$

Cotangent on the unit circle is the $x$-coordinate divided by the $y$-coordinate:

$$\cot\frac{3\pi}{4} = \frac{x}{y} = \frac{-0.7071}{0.7071} = -1$$

The $x$ is negative and the $y$ is positive, so their ratio is negative, which is the unit-circle way of seeing the same sign the ASTC rule gave. This is the double anchor Bhanzu insists on: cotangent as the shadow ratio of a right triangle, and cotangent as $x/y$ on the circle, are the same number viewed two ways.

Here is how $\frac{3\pi}{4}$ compares with its Quadrant II neighbours and the family of angles around it. Each cotangent is $x/y$, and each is negative because these angles all live where $x < 0$ and $y > 0$.

Table: Cotangent and its sibling ratios for the special angles from $\frac{\pi}{2}$ to $\pi$.

For the full grid of these values across all four quadrants, the trigonometric table and the trigonometric ratios of specific angles references lay them out end to end.

Why Is Cot 3pi/4 Equal To Negative One?

The answer $-1$ is not a coincidence of two square roots cancelling. It comes from two facts stacked on top of each other.

• The steepness is the same as at $45^\circ$. A terminal side at $135^\circ$ is the mirror image of one at $45^\circ$, reflected across the vertical axis. The triangle it makes with the $x$-axis has legs of equal length, so the ratio of horizontal to vertical is $1$ in size, exactly as at $\frac{\pi}{4}$.

• The direction flips the sign. In Quadrant II the horizontal leg points left ($x$ is negative) while the vertical leg points up ($y$ is positive). Cotangent is horizontal over vertical, so one negative and one positive make the ratio negative.

Put the magnitude and the sign together and the value must be $-1$: size $1$ from the $45^\circ$ shape, minus sign from the second quadrant. This is why $\cot\frac{3\pi}{4}$ and $\cot\frac{\pi}{4}$ share a digit but differ in sign, and it is the same reasoning that makes every Quadrant II cotangent negative.

Who Discovered The Cotangent Function?

Cotangent did not begin as a line on a circle. It began as the length of a shadow. Ancient and medieval astronomers told the time by planting a vertical stick, called a gnomon, and measuring the shadow it cast. The ratio of the stick's height to its shadow length is exactly the cotangent of the sun's angle above the horizon, which is why the earliest cotangent tables were literally called shadow tables.

Two earlier figures built the tables Abu al-Wafa stood on:

• Hipparchus of Nicaea (c. 190–120 BCE, Greece) compiled the first known table of chords, the ancestor of every later trigonometric table.

• Aryabhata (476–550 CE, India) produced an early sine table (he called the quantity jya), the tradition that later gave us the words "sine" and, through it, the reciprocal ratios.

Where Is Cot 3pi/4 Used In The Real World?

A specific value like $\cot\frac{3\pi}{4} = -1$ shows up wherever a slant is measured against the horizontal and the direction of that slant carries a sign.

• Surveying and shadows: the original use survives in surveying, where the height of an object is found from its shadow, a direct cotangent ratio measured off the angle of the sun.

• Ramps, roofs, and grades: the cotangent of an incline is its "run over rise," the horizontal distance covered per unit of climb, which engineers use to specify gentle slopes for access ramps and drainage.

• Alternating current: in electrical engineering the cotangent of a phase angle relates the reactive and resistive parts of a circuit, and the sign tells you which way the phase leans.

• Computer graphics: perspective-projection matrices are built from the cotangent of half the camera's field-of-view angle, converting a viewing angle into on-screen scaling.

• Navigation and astronomy: angle-of-elevation problems for stars and landmarks reduce to cotangent and tangent ratios, the same shadow-and-stick idea Abu al-Wafa used, now pointed at the sky.

Across all of them, one small fact, a $135^\circ$ slant giving $-1$, is a single entry in a toolkit that stretches from a builder's ramp to a game engine's camera.

What Are The Most Common Mistakes With Cot 3pi/4?

These four errors account for most wrong answers on this angle. Each one is a specific slip with a specific fix.

Dropping the negative sign.

Where it slips in:

A student finds the reference angle $\frac{\pi}{4}$, recalls $\cot\frac{\pi}{4} = 1$, and writes $\cot\frac{3\pi}{4} = 1$, forgetting the quadrant.

Don't do this:

Do not report the reference-angle value as the final answer. The reference angle fixes the size, never the sign.

The correct way:

Check the quadrant before writing the answer. $\frac{3\pi}{4}$ is in Quadrant II, where cotangent is negative, so the value is $-1$, not $+1$.

Leaving the calculator in the wrong angle mode.

Where it slips in:

A student types the angle as $3\pi/4 \approx 2.356$ while the calculator is set to degrees, or types $135$ while it is set to radians, and reads off a value that is nowhere near $-1$.

Don't do this:

Do not enter a radian angle in degree mode or the reverse. The two settings answer completely different questions.

The correct way:

Match the mode to the angle. Enter $135$ in degree mode or $3\pi/4$ in radian mode, and since most calculators lack a cotangent key, compute $1 \div \tan$ of the angle.

Misidentifying the reference angle.

Where it slips in:

A student subtracts from the wrong boundary, computing $\frac{3\pi}{4} - \frac{\pi}{2} = \frac{\pi}{4}$ by luck, or $\pi - \frac{3\pi}{4}$ incorrectly, and lands on a wrong acute angle for other cases.

Don't do this:

Do not guess which value to subtract from. In Quadrant II the reference angle is always $\pi$ minus the angle.

The correct way:

Use the quadrant rule: for a Quadrant II angle, reference angle $= \pi - \theta$. Here $\pi - \frac{3\pi}{4} = \frac{\pi}{4}$, which is correct.

Confusing cotangent with tangent.

Where it slips in:

A student treats $\cot\frac{3\pi}{4}$ as if it were $\tan\frac{3\pi}{4}$ without inverting, which happens to give $-1$ here and hides the error, then repeats the same shortcut on an angle where the two differ.

Don't do this:

Do not read cotangent off a tangent value without taking the reciprocal. At $\frac{3\pi}{4}$ they coincide only because $-1$ is its own reciprocal.

The correct way:

Apply $\cot\theta = \frac{1}{\tan\theta}$ every time. Test it on $\frac{2\pi}{3}$: there $\tan = -\sqrt{3}$ but $\cot = -\frac{\sqrt{3}}{3}$, so the reciprocal step genuinely matters.

Practice Problems On Cot 3pi/4

Work each one, then check against the answer beside it.

1. Evaluate $\cot\frac{3\pi}{4}$ using the reciprocal of tangent.
   (Answer: $\frac{1}{\tan\frac{3\pi}{4}} = \frac{1}{-1} = -1$.)

2. Find $\cot\frac{3\pi}{4} + \cot\frac{\pi}{4}$.
   (Answer: $-1 + 1 = 0$.)

3. Is $\cot\frac{3\pi}{4}$ positive or negative, and why?
   (Answer: negative, because $\frac{3\pi}{4}$ is in Quadrant II where cotangent is negative.)

4. Evaluate $\cot\frac{3\pi}{4} \times \sin\frac{3\pi}{4}$.
   (Answer: $-1 \times \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2} \approx -0.7071$, which is $\cos\frac{3\pi}{4}$.)

5. Convert $\frac{3\pi}{4}$ to degrees, then state its cotangent.
   (Answer: $135^\circ$, and $\cot 135^\circ = -1$.)

6. Using the reference angle and quadrant, find $\cot\frac{5\pi}{6}$.
   (Answer: reference angle $\frac{\pi}{6}$, Quadrant II, so $-\cot\frac{\pi}{6} = -\sqrt{3} \approx -1.7321$.)

Where Should You Go Next After Cot 3pi/4?

One angle opens onto the whole family of reciprocal ratios and special values.

1. Cosecant, secant, and cotangent functions. See how all three reciprocal ratios are defined and where each is positive or negative around the circle.

2. Unit circle with tangent. Build the reading you used here into a full picture of tangent and cotangent on the circle.

3. Trigonometric ratios of specific angles. Lock in the $30^\circ$, $45^\circ$, $60^\circ$ values that every quadrant reflects.

If your child is building this foundation, a live Bhanzu trainer teaches special-angle values starting from the reference-angle-and-quadrant reasoning above in the Bhanzu trigonometry program.