What Does Cos 7pi/4 Mean?
The four quadrants divide the plane by the axes: the fourth quadrant is the bottom-right, where the $x$-coordinate is positive and the $y$-coordinate is negative. The angle $\dfrac{7\pi}{4}$ lands here, between $\dfrac{3\pi}{2}$ and $2\pi$.
The reference angle is the acute angle between the terminal side and the $x$-axis. For $\dfrac{7\pi}{4}$ it is $2\pi - \dfrac{7\pi}{4} = \dfrac{\pi}{4}$. Cosine is the $x$-coordinate on the unit circle, and since $x$ is positive in the fourth quadrant, $\cos\left(\dfrac{7\pi}{4}\right)$ takes the positive $\dfrac{\pi}{4}$ value.
Where Does Cos 7pi/4 Show Up?
A point rotating counterclockwise reaches $\dfrac{7\pi}{4}$ just before completing a full turn, sitting low and to the right. Its cosine, the horizontal position, is still positive at $\dfrac{\sqrt{2}}{2}$, which is why a cosine wave has climbed most of the way back to its peak by this point.
The same value appears in any $45^\circ$ diagonal viewed from the fourth quadrant on the unit circle, where the horizontal and vertical reaches are equal in size.
Standard-Angle Reference Table
The angle $\dfrac{7\pi}{4}$ is $315^\circ$, one-eighth of a turn short of a full circle. Its cosine matches the $45^\circ$ family, shown here for the first quadrant.
Angle (radians) | Angle (degrees) | $\cos\theta$ (exact) | $\cos\theta$ (decimal) |
|---|---|---|---|
$\dfrac{\pi}{6}$ | $30^\circ$ | $\dfrac{\sqrt{3}}{2}$ | $0.8660$ |
$\dfrac{\pi}{4}$ | $45^\circ$ | $\dfrac{\sqrt{2}}{2}$ | $0.7071$ |
$\dfrac{\pi}{3}$ | $60^\circ$ | $\dfrac{1}{2}$ | $0.5000$ |
$\dfrac{7\pi}{4}$ | $315^\circ$ | $\dfrac{\sqrt{2}}{2}$ | $0.7071$ |
The value equals that of its reference-angle sibling cos pi/4, because $\dfrac{7\pi}{4}$ borrows the $\dfrac{\pi}{4}$ cosine and keeps it positive.
How Do You Find The Exact Value Of Cos 7pi/4?
Method 1: Reference angle plus quadrant sign.
Find the reference angle, take its known cosine, then fix the sign from the quadrant.
$$\text{Reference angle} = 2\pi - \frac{7\pi}{4} = \frac{\pi}{4}$$
The trigonometric ratios give $\cos\dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$. The fourth quadrant has a positive $x$, so the sign stays positive:
$$\cos\left(\frac{7\pi}{4}\right) = +\frac{\sqrt{2}}{2}$$
Method 2: The unit circle.
Sweep the radius by $\dfrac{7\pi}{4}$ and it lands at $\left(\dfrac{\sqrt{2}}{2}, -\dfrac{\sqrt{2}}{2}\right)$. Cosine is the $x$-coordinate:
$$\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}$$
The form $\dfrac{1}{\sqrt{2}}$ is the same number; rationalising the denominator gives the standard $\dfrac{\sqrt{2}}{2}$.
Examples Of Cos 7pi/4
Example 1
Evaluate $4\cos\left(\dfrac{7\pi}{4}\right)$.
$$4\cos\left(\frac{7\pi}{4}\right) = 4 \times \frac{\sqrt{2}}{2} = 2\sqrt{2} \approx 2.828$$
Example 2
A student places $\dfrac{7\pi}{4}$ in the third quadrant and writes $\cos\left(\dfrac{7\pi}{4}\right) = -\dfrac{\sqrt{2}}{2}$. What went wrong?
Wrong attempt. Assuming any large angle is "past halfway and so negative," the student records $-\dfrac{\sqrt{2}}{2}$.
That misreads the quadrant: $\dfrac{7\pi}{4}$ sits between $\dfrac{3\pi}{2}$ and $2\pi$, which is the fourth quadrant, not the third. In the fourth quadrant the $x$-coordinate is positive, so cosine cannot be negative here.
Correct. The reference angle is $\dfrac{\pi}{4}$ and the quadrant is fourth, so $\cos\left(\dfrac{7\pi}{4}\right) = +\dfrac{\sqrt{2}}{2}$.
Example 3
Convert $\dfrac{7\pi}{4}$ to degrees and confirm the value.
$$\frac{7\pi}{4} \times \frac{180^\circ}{\pi} = 315^\circ$$
Detail on this step lives at radians to degrees, and $\cos 315^\circ = \dfrac{\sqrt{2}}{2}$.
Example 4
Find $\sec\left(\dfrac{7\pi}{4}\right)$ from the cosine.
Secant is the reciprocal of cosine:
$$\sec\left(\frac{7\pi}{4}\right) = \frac{1}{\cos\left(\frac{7\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$$
The matching page sec 7pi/4 works this through in full.
Example 5
Verify $\cos^2\left(\dfrac{7\pi}{4}\right) + \sin^2\left(\dfrac{7\pi}{4}\right) = 1$, given $\sin\left(\dfrac{7\pi}{4}\right) = -\dfrac{\sqrt{2}}{2}$.
$$\left(\frac{\sqrt{2}}{2}\right)^2 + \left(-\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} + \frac{1}{2} = 1$$
Where Students Trip Up On Cos 7pi/4
Mistake 1: Putting 7pi/4 in the wrong quadrant
Where it slips in: Guessing the quadrant from the size of the angle instead of the boundaries.
Don't do this: Calling $\dfrac{7\pi}{4}$ a third-quadrant angle and making cosine negative.
The correct way: Compare against the quarter-turns: $\dfrac{7\pi}{4}$ is past $\dfrac{3\pi}{2}$ but short of $2\pi$, so it is fourth-quadrant, where cosine is positive.
Mistake 2: Computing the reference angle from the wrong axis
Where it slips in: Subtracting from $\pi$ instead of $2\pi$ for a fourth-quadrant angle.
Don't do this: Writing the reference angle as $\dfrac{7\pi}{4} - \pi = \dfrac{3\pi}{4}$.
The correct way: In the fourth quadrant the reference angle is $2\pi - \theta$, so $2\pi - \dfrac{7\pi}{4} = \dfrac{\pi}{4}$. The student who fixes the axis first stops producing reference angles larger than a right angle.
Mistake 3: Leaving cosine as 1 over root 2
Where it slips in: Stopping at the unrationalised form on a graded answer.
Don't do this: Writing $\cos\left(\dfrac{7\pi}{4}\right) = \dfrac{1}{\sqrt{2}}$ and treating it as final.
The correct way: Rationalise the denominator to the standard $\dfrac{\sqrt{2}}{2}$; the two forms are equal, but $\dfrac{\sqrt{2}}{2}$ is the expected exact answer.
Key Takeaways
Cos 7pi/4 equals $\dfrac{\sqrt{2}}{2}$, about $0.7071$, and it is positive.
$\dfrac{7\pi}{4}$ is $315^\circ$, a fourth-quadrant angle with reference angle $\dfrac{\pi}{4}$.
Reference angle sets the size, quadrant sets the sign: fourth-quadrant cosine stays positive.
Rationalise $\dfrac{1}{\sqrt{2}}$ to the standard $\dfrac{\sqrt{2}}{2}$ for the final answer.
To master reference angles alongside a teacher, explore Bhanzu's trigonometry tutor, its high school math tutor programme, or live math classes online.
Practice These Before Moving On
Evaluate $2\cos\left(\dfrac{7\pi}{4}\right) - \sqrt{2}$.
State the reference angle and quadrant of $\dfrac{7\pi}{4}$, then give its cosine.
Find $\cos\left(\dfrac{5\pi}{4}\right)$ and explain why its sign differs from $\cos\left(\dfrac{7\pi}{4}\right)$.
Want a live Bhanzu trainer to walk through more cos 7pi/4 problems? Book a free demo class.
Read More
Cos 45 Degrees — the first-quadrant angle that sets this reference value.
Cos 2pi — the full-turn value just past $\dfrac{7\pi}{4}$.
Cot 7pi/4 — the same angle through the cotangent ratio.
Sin, Cos, Tan — the three ratios defined together.
Trigonometric Ratios in Radians — reading the standard angles in radian form.
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