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