Cos 7pi/4 : Exact Value, root 2 over 2, on the Unit Circle

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.

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

1. Evaluate $2\cos\left(\dfrac{7\pi}{4}\right) - \sqrt{2}$.

2. State the reference angle and quadrant of $\dfrac{7\pi}{4}$, then give its cosine.

3. 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.