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

The value of $\sec\frac{7\pi}{4}$ is $\sqrt{2}$.

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

Neighbouring angles in both notations, with their secant values.

What Secant of an Angle Means

Secant is the reciprocal of cosine: $\sec\theta = \frac{1}{\cos\theta}$. 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 $\cos\frac{7\pi}{4} = \frac{\sqrt{2}}{2}$. Then flip it:

$$\sec\frac{7\pi}{4} = \frac{1}{\cos\frac{7\pi}{4}} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$$

Final answer: $\sqrt{2}$.

Method 2: Convert radians to degrees

$$\frac{7\pi}{4} \times \frac{180°}{\pi} = \frac{7 \times 180°}{4} = 315°$$

So $\sec\frac{7\pi}{4} = \sec 315°$. Going back, 315° converts with $\frac{\pi}{180°}$. 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 $2\pi$ minus the angle.

$$2\pi - \frac{7\pi}{4} = \frac{\pi}{4}$$

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

$$\sec\frac{7\pi}{4} = +\sec\frac{\pi}{4} = \sqrt{2}$$

Final answer: $\sqrt{2}$.

The same 315° terminal point gives cot 7pi/4, which equals −1; comparing the two shows how the positive secant and negative cotangent both come from one point on the circle.

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 $\sin\frac{7\pi}{4} = -\frac{\sqrt{2}}{2}$ where secant is asked.

The correct way: Secant is the reciprocal of cosine, not the reciprocal of sine. Anchor it: $\sec \leftrightarrow \cos$, $\csc \leftrightarrow \sin$. The reciprocal $\frac{1}{\cos\theta}$ and the inverse $\cos^{-1}\theta$ are also different ideas, so don't swap them when checking.

Mistake 2: Forgetting to rationalise the reciprocal

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

Don't do this: Leave $\sec\frac{7\pi}{4} = \frac{2}{\sqrt{2}}$ as the final form.

The correct way: Simplify $\frac{2}{\sqrt{2}} = \sqrt{2}$ by multiplying top and bottom by $\sqrt{2}$. The clean form $\sqrt{2}$ is the standard answer.

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 $\sec\frac{7\pi}{4} = -\sqrt{2}$.

The correct way: Secant copies cosine's sign, and cosine is the $x$-coordinate. At 315° the $x$-coordinate is positive, so secant is positive.

To practise reciprocal functions like secant with a live teacher, Bhanzu's trigonometry tutor and math tutoring walk through cosine first and the reciprocal second, the order that keeps signs straight.

Read More

• Reciprocal identities for sec, cosec, and cot

• Trigonometric ratios and their definitions

• Sin Cos Tan — the parent ratios

• The trigonometric table of standard values

• Trigonometric functions and their definitions