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

Trigonometry

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

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.

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 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 and math tutoring 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}.