Cos 5pi/4 - Exact Value -√2/2 and How to Find It

What Does Cos 5pi/4 Mean?

On the unit circle, the cosine of an angle is the $x$-coordinate of the point where the angle's radius meets the circle. A quadrant is one of the four regions the axes cut the plane into, numbered anticlockwise from the top right.

The angle $\dfrac{5\pi}{4}$ is measured anticlockwise from the positive $x$-axis and stops in Quadrant III, the bottom-left region. Its terminal point is $\left(-\dfrac{\sqrt{2}}{2}, -\dfrac{\sqrt{2}}{2}\right)$, so the $x$-coordinate, and therefore the cosine, is $-\dfrac{\sqrt{2}}{2}$. Points in Quadrant III have both coordinates negative, which is why cosine and sine are both negative here.

Where Does Cos 5pi/4 Show Up?

The angle $\dfrac{5\pi}{4}$ points to $225^\circ$, which is $45^\circ$ below the negative $x$-axis, straight into the bottom-left. A vector pointing southwest, a force acting down and to the left, or a compass bearing in that direction all have a horizontal component read from $\cos\dfrac{5\pi}{4} = -\dfrac{\sqrt{2}}{2}$. The value is negative because the point has swung past the vertical into the left half of the circle.

Standard-Angle Reference Table

The value $\dfrac{\sqrt{2}}{2}$ and its negative appear at four angles that share the reference angle $\dfrac{\pi}{4}$. The quadrant decides the sign.

All four have the same reference angle, so their cosines share the magnitude $\dfrac{\sqrt{2}}{2}$. Since $\dfrac{5\pi}{4}$ sits in Quadrant III, its cosine is negative, matching cos 135 degrees in value while sitting one quadrant further round.

How Do You Find The Exact Value Of Cos 5pi/4?

Three routes all give $-\dfrac{\sqrt{2}}{2}$.

Method 1: Reference angle and quadrant sign.

A reference angle is the acute angle between the terminal side and the $x$-axis. For a Quadrant III angle, subtract $\pi$:

$$\frac{5\pi}{4} - \pi = \frac{5\pi - 4\pi}{4} = \frac{\pi}{4}$$

Cosine is negative in Quadrant III, so:

$$\cos\frac{5\pi}{4} = -\cos\frac{\pi}{4} = -\frac{\sqrt{2}}{2}$$

Method 2: The unit circle.

Convert with the radians-to-degrees rule: $\dfrac{5\pi}{4} = 225^\circ$. The terminal point at $225^\circ$ is $\left(-\dfrac{\sqrt{2}}{2}, -\dfrac{\sqrt{2}}{2}\right)$, so

$$\cos\frac{5\pi}{4} = x\text{-coordinate} = -\frac{\sqrt{2}}{2}$$

Method 3: From the reference-angle twin.

The magnitude comes straight from cos π/4, which is $\dfrac{\sqrt{2}}{2}$ or equivalently $\dfrac{1}{\sqrt{2}}$. The rationalised form $\dfrac{\sqrt{2}}{2}$ is the standard way to write it, and Quadrant III attaches the minus sign:

$$\cos\frac{5\pi}{4} = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}$$

Examples Of Cos 5pi/4

Example 1

Evaluate $4\sqrt{2},\cos\dfrac{5\pi}{4}$.

$$4\sqrt{2} \times \left(-\frac{\sqrt{2}}{2}\right) = -\frac{4 \times 2}{2} = -4$$

Example 2

Find $\cos\dfrac{5\pi}{4}$ using the reference angle.

Wrong attempt. A student finds the reference angle $\dfrac{\pi}{4}$, reads $\cos\dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$, and writes $\cos\dfrac{5\pi}{4} = \dfrac{\sqrt{2}}{2}$.

That skips the quadrant step. The reference angle only fixes the size of the value, not its sign, and $\dfrac{5\pi}{4}$ is in Quadrant III where the $x$-coordinate is negative.

Correct. Apply the Quadrant III sign to the magnitude: $\cos\dfrac{5\pi}{4} = -\dfrac{\sqrt{2}}{2}$.

Example 3

Compare $\cos\dfrac{5\pi}{4}$ with $\cos\dfrac{7\pi}{4}$.

Both share the reference angle $\dfrac{\pi}{4}$, so both have magnitude $\dfrac{\sqrt{2}}{2}$. But $\dfrac{7\pi}{4}$ is in Quadrant IV where cosine is positive:

$$\cos\frac{5\pi}{4} = -\frac{\sqrt{2}}{2}, \qquad \cos\frac{7\pi}{4} = \frac{\sqrt{2}}{2}$$

Example 4

Evaluate $\cos\dfrac{5\pi}{4} + \sin\dfrac{5\pi}{4}$.

Both coordinates are $-\dfrac{\sqrt{2}}{2}$ in Quadrant III:

$$-\frac{\sqrt{2}}{2} + \left(-\frac{\sqrt{2}}{2}\right) = -\sqrt{2} \approx -1.414$$

Example 5

Find $\sec\dfrac{5\pi}{4}$.

Secant is the reciprocal of cosine:

$$\sec\frac{5\pi}{4} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2}$$

Where Students Trip Up On Cos 5pi/4

Mistake 1: Forgetting the Quadrant III minus sign

Where it slips in: Finding the reference angle, reading the positive value, and stopping there.

Don't do this: Writing $\cos\dfrac{5\pi}{4} = \dfrac{\sqrt{2}}{2}$. The reference-angle value is always positive, and it is easy to hand in that number without applying the quadrant sign.

The correct way: In Quadrant III both coordinates are negative, so cosine is negative: $\cos\dfrac{5\pi}{4} = -\dfrac{\sqrt{2}}{2}$. Always pair the reference angle with a quadrant-sign check.

Mistake 2: Using the wrong reference angle

Where it slips in: Subtracting from $2\pi$ or from $\dfrac{\pi}{2}$ instead of from $\pi$.

Don't do this: Computing $2\pi - \dfrac{5\pi}{4} = \dfrac{3\pi}{4}$ and treating that as the reference angle.

The correct way: For a Quadrant III angle, the reference angle is the angle minus $\pi$: $\dfrac{5\pi}{4} - \pi = \dfrac{\pi}{4}$.

Mistake 3: Leaving the value unrationalised

Where it slips in: Stopping at $-\dfrac{1}{\sqrt{2}}$ when a rationalised answer is expected.

Don't do this: Treating $-\dfrac{1}{\sqrt{2}}$ and $-\dfrac{\sqrt{2}}{2}$ as if only one is acceptable when the form matters.

The correct way: They are equal, but the rationalised $-\dfrac{\sqrt{2}}{2}$ is the standard form. Bearings and direction vectors carry this same idea: $225^\circ$ points southwest, and reporting the horizontal component as a clean $-\dfrac{\sqrt{2}}{2}$ keeps later calculations consistent.

Key Takeaways

• Cos 5pi/4 equals $-\dfrac{\sqrt{2}}{2}$, about $-0.7071$ - negative because $\dfrac{5\pi}{4}$ lies in Quadrant III.

• The reference angle is $\dfrac{\pi}{4}$, so the magnitude matches $\cos 45^\circ = \dfrac{\sqrt{2}}{2}$; the quadrant supplies the minus sign.

• In degrees, $\cos\dfrac{5\pi}{4} = \cos 225^\circ = -\dfrac{\sqrt{2}}{2}$, and the terminal point is $\left(-\dfrac{\sqrt{2}}{2}, -\dfrac{\sqrt{2}}{2}\right)$.

• The most common slip is reporting the positive reference-angle value and forgetting the Quadrant III sign.

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Practice These Before Moving On

1. Evaluate $2\cos\dfrac{5\pi}{4} + \sec\dfrac{5\pi}{4}$.

2. Without a calculator, decide the sign of $\cos\dfrac{3\pi}{4}$ and give its value.

3. Rewrite $-\dfrac{1}{\sqrt{2}}$ in rationalised form and confirm it equals $\cos\dfrac{5\pi}{4}$.

Want a live Bhanzu trainer to walk through quadrant signs and rationalised forms? Book a free demo class.

Read More

• Cos 45 Degrees — the reference angle in degree form, equal to $\dfrac{\sqrt{2}}{2}$.

• Cos 3pi/4 — the Quadrant II angle with the same negative value.

• Cos 7pi/4 — the Quadrant IV partner where the sign flips to positive.

• Cosine function — how cosine behaves across all four quadrants.

• Trigonometric ratios of specific angles — the full standard-angle set.