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# Sec 7pi/4 — Exact Value, Unit Circle, Methods

[Trigonometry](/content/tag/trigonometry/index.html)

TL;DR

Sec 7pi/4 is √2 (about 1.4142), because secant is the reciprocal of cosine and cos⁡7π/4=\frac{\sqrt{2}}{2} at 315° in the fourth quadrant, where cosine is positive. This article finds the value through the reciprocal identity, the degree conversion, and the π/4 reference angle.

The value of sec⁡7π/4 is 2\sqrt{2}.

> **Quick Answer:**  
> **Result:** sec(7π/4) = √2 ≈ 1.4142  
> **Notation:** exact surd √2  
> **Method shown:** reciprocal of cosine + reference angle on the unit circle  
> **Degree equivalent:** sec 315°  
> **Sign:** positive (fourth quadrant)

## Quick Reference Table

| Angle (radians)   | Angle (degrees) | Quadrant | sec⁡ \sec value         |
| ------------------- | ---------------- | -------- | --------------------- |
| \frac{\pi}{4}     | 45°              | I        | 2\sqrt{2}             |
| \frac{3\pi}{4}   | 135°             | II       | −2−\sqrt{2}           |
| \pi                | 180°             | —        | −1                     |
| \frac{5\pi}{4}   | 225°             | III      | −2−\sqrt{2}           |
| \frac{3\pi}{2}   | 270°             | —        | undefined              |
| \frac{7\pi}{4}   | 315°             | IV       | 2\sqrt{2}             |

## What Secant of an Angle Means

Secant is the reciprocal of cosine: sec⁡θ=1cos⁡θ. Because it is built from cosine, it shares cosine's sign exactly. On the unit circle, where cosine is the x-coordinate of the terminal point, secant is one divided by that x-coordinate.

A **quadrant** is one of the four regions the axes divide the plane into, numbered I to IV anticlockwise. Cosine — and therefore secant — is positive in quadrants I and IV and negative in II and III. The angle 7π/4 sits in the fourth quadrant, so its secant is positive. For more on the function itself, see the [secant function](/content/math/trigonometry/secant-function/index.html).

## Methods to Find Sec 7pi/4

How do you find sec 7pi/4 without a calculator? Start from cosine; secant follows.

### **Method 1: Reciprocal of cosine (the direct route)**

First find the cosine. At 315° the unit-circle point is \left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right), so cos⁡7π/4=\frac{\sqrt{2}}{2}. Then flip it:

sec⁡7π/4=\frac{1}{\cos⁡7π/4}=\frac{1}{\frac{\sqrt{2}}{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}.

**Final answer:** 2\sqrt{2}.

### **Method 2: Convert radians to degrees**

7π/4×\frac{180°}{\pi}=\frac{7×180°}{4}=315°.

So sec⁡7π/4=sec⁡315°. The [radian-to-degree conversion](/content/math/trigonometry/1-radian-to-degrees/index.html) shows where that factor comes from.

**Final answer:** 315°, then apply the reciprocal as in Method 1.

### **Method 3: Reference angle**

For a fourth-quadrant angle the reference angle is 2π minus the angle.

2π−7π/4=\frac{\pi}{4}.

The reference angle is \frac{\pi}{4} (45°), and sec⁡\frac{\pi}{4}=2. Quadrant IV keeps secant positive, so:

sec⁡7π/4=+sec⁡\frac{\pi}{4}=2.

**Final answer:** 2\sqrt{2}.

## Common Mistakes of Sec 7pi/4

### **Mistake 1: Confusing secant with sine**

**Where it slips in:** The lookalike names sec⁡ and sin⁡, especially when reading fast.

**Don't do this:** Use sin⁡7π/4=−\frac{\sqrt{2}}{2} where secant is asked.

**The correct way:** Secant is the reciprocal of **cosine**, not the reciprocal of sine.

### **Mistake 2: Forgetting to rationalize the reciprocal**

**Where it slips in:** Stopping at \frac{2}{\sqrt{2}} instead of simplifying.

**Don't do this:** Leave sec⁡7π/4=\frac{2}{\sqrt{2}} as the final form.

**The correct way:** Simplify to 2\sqrt{2}.

### **Mistake 3: Assigning a negative sign because the angle is "near the bottom"**

**Where it slips in:** Reading 315° as a downward direction and assuming a negative output.

**Don't do this:** Write sec⁡7π/4=−2.

**The correct way:** Secant copies cosine's sign, and cosine is positive at 315°.

To practice reciprocal functions like secant with a live teacher, Bhanzu's [trigonometry tutor](/content/math/tutor/trigonometry/index.html) and [math tutoring](/content/math/tutoring/index.html) walk through cosine first and the reciprocal second.

## Frequently Asked Questions

**Is sec 7pi/4 positive or negative?**

Positive. Secant shares cosine's sign, and cosine — the x-coordinate at 315° — is positive in the fourth quadrant.

**What is sec 7pi/4 as a decimal?**

About 1.4142. The exact form 2\sqrt{2} is preferred because it never rounds.

**How do you get sec 7pi/4 from cos 7pi/4?**

Take the reciprocal. With cos⁡7π/4=\frac{\sqrt{2}}{2}, you have sec⁡7π/4=2\sqrt{2}.

**What is the reference angle for 7pi/4?**

\frac{\pi}{4}, or 45°, measured from the positive x-axis.

**Does sec 7pi/4 equal sec 315°?**

Yes. 7π/4 radians is exactly 315°, so the secant is the same value, 2\sqrt{2}.