Cos 3pi/4 - Exact Value -√2/2 and the Reference Angle

What Does Cos 3pi/4 Mean?

Cosine is one of the three core trigonometric ratios, and on the unit circle it is the $x$-coordinate of the point where the angle's radius meets the circle. So $\cos \dfrac{3\pi}{4}$ asks: after rotating $\dfrac{3\pi}{4}$ radians from the positive $x$-axis, what is the $x$-coordinate?

A radian measure of $\dfrac{3\pi}{4}$ is three-quarters of the way to a half-turn, landing in the upper-left quadrant at $\left(-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)$. The $x$-coordinate there is negative, so the cosine is $-\dfrac{\sqrt{2}}{2}$.

Where Does Cos 3pi/4 Show Up?

The angle $\dfrac{3\pi}{4}$ is a natural "diagonal into the back-left" direction, so it appears whenever a vector or force points up and to the left at $45^\circ$ above the horizontal. Its cosine, $-\dfrac{\sqrt{2}}{2}$, gives the horizontal component of a unit push in that direction.

In wave problems and phasor diagrams, $\dfrac{3\pi}{4}$ is a common phase shift, and the cosine function evaluated there returns this exact negative fraction. The value is one of the standard second-quadrant results catalogued alongside the other special right triangle angles.

Standard-Angle Reference Table

The angle $\dfrac{3\pi}{4}$ sits in the second quadrant. It shares a reference angle with $\dfrac{\pi}{4}$, so its cosine has the same size but the opposite sign.

Read across the second row of interest: $\dfrac{\pi}{4}$ and $\dfrac{3\pi}{4}$ carry the same $\dfrac{\sqrt{2}}{2}$ magnitude, but the Quadrant II angle flips the sign to negative.

How Do You Find The Exact Value Of Cos 3pi/4?

The reference-angle method is the cleanest route, and the unit circle confirms it.

Method 1: Reference angle plus quadrant sign.

First find the reference angle - the acute angle between the terminal side and the $x$-axis. For $\dfrac{3\pi}{4}$ (that is $135^\circ$), the reference angle is $\pi - \dfrac{3\pi}{4} = \dfrac{\pi}{4}$.

The cosine of the reference angle is $\cos \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$, the value shared with cos 45 degrees. Because $\dfrac{3\pi}{4}$ sits in Quadrant II, where $x$ is negative and cosine is negative:

$$\cos \frac{3\pi}{4} = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2}$$

Method 2: The unit circle.

Rotate the radius $\dfrac{3\pi}{4}$ radians into the upper-left quadrant. The point lands at $\left(-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)$.

$$\cos \frac{3\pi}{4} = x\text{-coordinate} = -\frac{\sqrt{2}}{2}$$

Method 3: In degrees.

Convert: $\dfrac{3\pi}{4}$ radians $= \dfrac{3}{4} \times 180^\circ = 135^\circ$, so $\cos \dfrac{3\pi}{4} = \cos 135^\circ = -\dfrac{\sqrt{2}}{2}$, matching the value at cos 135 degrees.

Examples Of Cos 3pi/4

Example 1

Evaluate $4\cos \dfrac{3\pi}{4}$.

$$4\cos \frac{3\pi}{4} = 4 \times \left(-\frac{\sqrt{2}}{2}\right) = -2\sqrt{2} \approx -2.828$$

Example 2

Find $\cos \dfrac{3\pi}{4}$ using its reference angle.

Wrong attempt. A student finds the reference-angle cosine $\dfrac{\sqrt{2}}{2}$ and stops there, writing $\cos \dfrac{3\pi}{4} = \dfrac{\sqrt{2}}{2}$.

That breaks the sign check: $\dfrac{3\pi}{4}$ is in Quadrant II, where the $x$-coordinate is to the left of the origin and must be negative. A positive answer places the point in the wrong half of the circle.

Correct. Apply the Quadrant II sign: $\cos \dfrac{3\pi}{4} = -\dfrac{\sqrt{2}}{2}$. The reference angle sets the size; the quadrant sets the sign.

Example 3

Verify the identity $\cos^2 \dfrac{3\pi}{4} + \sin^2 \dfrac{3\pi}{4} = 1$.

$$\left(-\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} + \frac{1}{2} = 1$$

The Pythagorean identity holds, confirming the pair of coordinates is genuinely on the unit circle.

Example 4

Show that $\cos \dfrac{3\pi}{4} = -\cos \dfrac{\pi}{4}$.

The reference angle of $\dfrac{3\pi}{4}$ is $\dfrac{\pi}{4}$, and the Quadrant II sign is negative, so:

$$\cos \frac{3\pi}{4} = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2}$$

This is the supplementary-angle relationship $\cos(\pi - \theta) = -\cos\theta$ with $\theta = \dfrac{\pi}{4}$.

Example 5

A unit force acts at $\dfrac{3\pi}{4}$ radians above the positive $x$-axis. Find its horizontal component.

The horizontal component is $\cos \dfrac{3\pi}{4}$.

$$\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \approx -0.7071$$

The negative sign shows the force pushes to the left.

Where Students Trip Up On Cos 3pi/4

Mistake 1: Dropping the negative sign

Where it slips in: The reference-angle method, when a reader computes the size but forgets the quadrant sign.

Don't do this: Writing $\cos \dfrac{3\pi}{4} = \dfrac{\sqrt{2}}{2}$. That is the first-quadrant value at $\dfrac{\pi}{4}$, not the Quadrant II value.

The correct way: In Quadrant II the $x$-coordinate is negative, so $\cos \dfrac{3\pi}{4} = -\dfrac{\sqrt{2}}{2}$. Students who race to the reference angle without pausing on the quadrant sign lose the negative every time.

Mistake 2: Using the wrong reference angle

Where it slips in: Finding the reference angle, when a reader measures from the $y$-axis or subtracts from the wrong base.

Don't do this: Writing the reference angle as $\dfrac{3\pi}{4}$ itself, or as $\dfrac{\pi}{2} - \dfrac{3\pi}{4}$, which gives a negative angle.

The correct way: In Quadrant II, the reference angle is $\pi - \theta$. Here $\pi - \dfrac{3\pi}{4} = \dfrac{\pi}{4}$, an acute angle measured to the negative $x$-axis.

Mistake 3: Leaving the answer unrationalised or as a rounded decimal

Where it slips in: Calculator-first solving, where the screen reads $-0.707$ and the student copies it.

Don't do this: Writing $\cos \dfrac{3\pi}{4} = -0.707$ on a problem that asks for the exact value, or leaving it as $-\dfrac{1}{\sqrt{2}}$ with an unrationalised denominator.

The correct way: Give the standard rationalised form $-\dfrac{\sqrt{2}}{2}$. The decimal $-0.7071$ is an approximation; $-\dfrac{\sqrt{2}}{2}$ is the value.

Key Takeaways

• Cos 3pi/4 equals $-\dfrac{\sqrt{2}}{2}$, approximately $-0.7071$ — an exact value because $\dfrac{3\pi}{4}$ is built from the $\dfrac{\pi}{4}$ reference angle.

• The reference angle $\dfrac{\pi}{4}$ sets the size; the Quadrant II position sets the negative sign.

• In degrees, $\cos \dfrac{3\pi}{4} = \cos 135^\circ$, and the standard rationalised form is $-\dfrac{\sqrt{2}}{2}$.

• To take reference-angle work further with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or live math classes online.

Practice These Before Moving On

1. Evaluate $2\cos \dfrac{3\pi}{4} + \sin \dfrac{3\pi}{4}$.

2. Show that $\cos \dfrac{3\pi}{4} = \cos \dfrac{5\pi}{4}$ is false, and state the correct value of each.

3. A force of $10$ N acts at $\dfrac{3\pi}{4}$ radians. Find its horizontal component using $\cos \dfrac{3\pi}{4}$.

Want a live Bhanzu trainer to walk through more cos 3pi/4 problems? Book a free demo class.

Read More

• Cos 2pi/3 — another second-quadrant angle with an exact fraction value.

• Cos 120 Degrees — the degree form of a nearby Quadrant II angle.

• Cos pi/4 — the first-quadrant twin that supplies this angle's reference value.

• Trigonometric Table — sine, cosine, and tangent for every standard angle in one chart.