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Cot 7pi/4 — Exact Value, Unit Circle, Methods

Trigonometry

TL;DR

Cot 7pi/4 is −1, because 7π/4 lands at 315° in the fourth quadrant where cotangent is negative and the reference angle is π/4. This article finds the value through the degree conversion, the reference angle, and the cosine-over-sine quotient.

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Bhanzu Team Last updated on July 15, 2026 3 min read

What Cotangent of an Angle Means

Cotangent is the ratio of cosine to sine: cot⁡θ=cos⁡θsin⁡θ\cot\theta = \frac{\cos\theta}{\sin\theta}cotθ=sinθcosθ​, equivalently 1tan⁡θ\frac{1}{\tan\theta}tanθ1​. On the unit circle it is the xxx-coordinate divided by the yyy-coordinate of the terminal point.

A quadrant is one of the four regions the axes split the plane into, numbered I to IV anticlockwise. Cotangent follows tangent's sign: positive in quadrants I and III, negative in II and IV. The angle 7π/4 sits in the fourth quadrant, so its cotangent is negative from the start. The reciprocal link is one of the standard reciprocal identities.

Methods to Find Cot 7pi/4

How do you evaluate cot 7pi/4 by hand? Three routes agree on −1.

Method 1: Convert radians to degrees

7π4×180°π=7×180°4=315°\frac{7\pi}{4} \times \frac{180°}{\pi} = \frac{7 \times 180°}{4} = 315°47π​×π180°​=47×180°​=315°

So cot⁡7π4=cot⁡315°\cot\frac{7\pi}{4} = \cot 315°cot47π​=cot315°, and 315° converts back with π180°\frac{\pi}{180°}180°π​. The radian-to-degree conversion is the bridge if you prefer degrees.

Final answer: 315°315°315°.

Method 2: Reference angle

For a fourth-quadrant angle the reference angle is 2π2\pi2π minus the angle.

2π−7π4=8π−7π4=π42\pi - \frac{7\pi}{4} = \frac{8\pi - 7\pi}{4} = \frac{\pi}{4}2π−47π​=48π−7π​=4π​

The reference angle is π4\frac{\pi}{4}4π​ (45°), and cot⁡π4=1\cot\frac{\pi}{4} = 1cot4π​=1.

Quadrant IV makes cotangent negative, so:

cot⁡7π4=−cot⁡π4=−1\cot\frac{7\pi}{4} = -\cot\frac{\pi}{4} = -1cot47π​=−cot4π​=−1

Final answer: −1-1−1.

Method 3: As the reciprocal of tangent (cos over sin)

At 315° the unit-circle point is (22,−22)\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)(22​​,−22​​). Cotangent is xxx over yyy:

cot⁡7π4=cos⁡7π4sin⁡7π4=22−22=−1\cot\frac{7\pi}{4} = \frac{\cos\frac{7\pi}{4}}{\sin\frac{7\pi}{4}} = \frac{\tfrac{\sqrt{2}}{2}}{-\tfrac{\sqrt{2}}{2}} = -1cot47π​=sin47π​cos47π​​=−22​​22​​​=−1

Since tan⁡7π4=−1\tan\frac{7\pi}{4} = -1tan47π​=−1, its reciprocal 1−1\frac{1}{-1}−11​ is again −1-1−1 — a tidy check.

Final answer: −1-1−1.

The same 315° angle drives sec 7pi/4, which equals √2 — the two pages share a terminal point but ask for different ratios, so reading them together cements the quadrant-IV picture.

Common Mistakes of Cot 7pi/4

Mistake 1: Treating cot as 1/tan when tan is zero or undefined

Where it slips in: Reusing the cot⁡θ=1tan⁡θ\cot\theta = \frac{1}{\tan\theta}cotθ=tanθ1​ shortcut at axis angles.

Don't do this: Write cot⁡7π4\cot\frac{7\pi}{4}cot47π​ confidently from the reciprocal but then apply the same move at 270°, where tangent is undefined.

The correct way: At 7π/4 the reciprocal is fine (tan⁡=−1\tan = -1tan=−1). But fall back on cos⁡θsin⁡θ\frac{\cos\theta}{\sin\theta}sinθcosθ​ at axis angles — that definition handles the cases where tangent breaks down.

Mistake 2: Dropping the negative sign

Where it slips in: After the reference value cot⁡π4=1\cot\frac{\pi}{4} = 1cot4π​=1, which is positive.

Don't do this: Report cot⁡7π4=1\cot\frac{7\pi}{4} = 1cot47π​=1.

The correct way: The reference angle sets the size; quadrant IV makes the sign negative because sine is negative there while cosine is positive.

Mistake 3: Using π instead of 2π for the fourth-quadrant reference

Where it slips in: Carrying over the third-quadrant rule (subtract π) into quadrant IV.

Don't do this: Compute 7π4−π=3π4\frac{7\pi}{4} - \pi = \frac{3\pi}{4}47π​−π=43π​ and call it the reference angle.

The correct way: In quadrant IV the reference angle is 2π−angle=2π−7π4=π42\pi - \text{angle} = 2\pi - \frac{7\pi}{4} = \frac{\pi}{4}2π−angle=2π−47π​=4π​.

For step-by-step practice on cotangent across all four quadrants with a teacher, Bhanzu's trigonometry tutor and math classes online build the unit circle one quadrant at a time.