Cos 135 Degrees - Value −√2/2 Explained

The value of cos 135 degrees is $-\frac{\sqrt{2}}{2}$, or approximately $-0.7071$.

Quick Answer:

• Result: $\cos 135° = -\dfrac{\sqrt{2}}{2}$

• Decimal: $\approx -0.7071$

• In radians: $\cos\left(\dfrac{3\pi}{4}\right) = -\dfrac{\sqrt{2}}{2}$

• Exact form: $-\dfrac{\sqrt{2}}{2}$, equivalently $-\dfrac{1}{\sqrt{2}}$ (a standard angle — exact, not rounded)

• Methods shown: reference angle (Quadrant II) · unit circle x-coordinate · supplementary identity

Standard-Angle Cosine Reference Table

One hundred thirty-five degrees is a standard angle, so its cosine has an exact radical form. Here are the common angles across Quadrants I and II, in degrees and radians.

Cos 135° and cos 45° carry the same magnitude, $\frac{\sqrt{2}}{2}$, with opposite signs — because $45°$ is the reference angle of $135°$, and cosine turns negative once you cross $90°$ into Quadrant II.

Where Cos 135 Degrees Shows Up

A 135° turn is a "half-back" direction — it points up and to the left, exactly between straight-left and straight-up. In 2D game physics and robotics, a movement vector at $135°$ has equal-magnitude components ($-\frac{\sqrt{2}}{2}$ across, $+\frac{\sqrt{2}}{2}$ up), so a sprite moving diagonally back-left uses $\cos 135°$ for its horizontal speed.

The same value appears in the trigonometric ratios behind a 45° roof rafter measured from the far side, and in alternating-current phase math where signals sit a quarter-turn-plus apart. The negative value follows directly from the unit circle, where a Quadrant II point has a negative $x$-coordinate.

What Cos 135 Degrees Means

On the unit circle — a circle of radius $1$ centred at the origin — the cosine of an angle is the $x$-coordinate of the point where the angle's radius meets the circle. Rotating $135°$ counterclockwise lands in the upper-left region (Quadrant II) at $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, whose negative $x$-coordinate gives $\cos 135° = -\frac{\sqrt{2}}{2}$.

The right-triangle definition of cosine — adjacent over hypotenuse — handles only acute angles, so the unit circle is the working definition past $90°$. The triangle returns through the reference angle, which for $135°$ is $180° - 135° = 45°$ — and $45°$ is the angle of an isosceles right triangle, where adjacent and opposite are equal.

How Do You Find the Value of Cos 135 Degrees?

Find the magnitude from the reference angle, then attach the quadrant's sign. The three routes below all give $-\frac{\sqrt{2}}{2}$.

Method 1: Reference angle

The reference angle for $135°$ is the acute angle to the negative $x$-axis:

$$180° - 135° = 45°$$

The magnitude matches: $\cos 45° = \frac{\sqrt{2}}{2}$. Since $135°$ is in Quadrant II, cosine is negative:

$$\cos 135° = -\cos 45° = -\frac{\sqrt{2}}{2}$$

Method 2: Unit circle

Rotate the unit radius $135°$ counterclockwise. It lands at $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

$$\cos 135° = x\text{-coordinate} = -\frac{\sqrt{2}}{2}$$

Method 3: Supplementary-angle identity

Using $\cos(180° - \theta) = -\cos\theta$:

$$\cos 135° = \cos(180° - 45°) = -\cos 45° = -\frac{\sqrt{2}}{2}$$

A note on form: $-\frac{\sqrt{2}}{2}$ and $-\frac{1}{\sqrt{2}}$ are the same number, since multiplying $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ rationalizes the denominator. Most textbooks prefer the rationalized $-\frac{\sqrt{2}}{2}$.

Examples of Cos 135 Degrees

Example 1

Evaluate $2\cos 135°$.

$$2\cos 135° = 2 \times \left(-\frac{\sqrt{2}}{2}\right) = -\sqrt{2} \approx -1.414$$

Example 2

Find $\cos 135°$ from its reference angle.

Wrong attempt. A student writes the reference angle as $135°$ itself and reports $\cos 135° = \frac{\sqrt{2}}{2}$, positive.

That fails two checks: $135°$ is not acute, so it cannot be its own reference angle, and a Quadrant II angle must have a negative cosine.

Correct. Reduce first: reference angle $= 180° - 135° = 45°$, magnitude $\cos 45° = \frac{\sqrt{2}}{2}$, then apply the Quadrant II sign:

$$\cos 135° = -\frac{\sqrt{2}}{2}$$

Example 3

Evaluate $\cos 135° + \cos 45°$.

$$-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = 0$$

The reference-angle partners cancel across the $90°$ line.

Example 4

Verify $\cos^2 135° + \sin^2 135° = 1$, given $\sin 135° = \frac{\sqrt{2}}{2}$.

$$\left(-\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} + \frac{2}{4} = 1$$

The Pythagorean identity holds; squaring erases the sign.

Example 5

Express $135°$ in radians and evaluate $\cos\left(\frac{3\pi}{4}\right)$.

$135° = \frac{3\pi}{4}$ radians, so $\cos\left(\frac{3\pi}{4}\right) = \cos 135° = -\frac{\sqrt{2}}{2}$.

Common Confusions With Cos 135 Degrees

Mistake 1: Keeping the sign positive

Where it slips in: Lifting the magnitude from $\cos 45° = \frac{\sqrt{2}}{2}$ and forgetting the Quadrant II sign.

Don't do this: Writing $\cos 135° = \frac{\sqrt{2}}{2}$.

The correct way: Cosine is negative in Quadrant II, so $\cos 135° = -\frac{\sqrt{2}}{2}$.

Mistake 2: Mismatching the two exact forms

Where it slips in: Treating $-\frac{\sqrt{2}}{2}$ and $-\frac{1}{\sqrt{2}}$ as different values.

Don't do this: Marking $-\frac{1}{\sqrt{2}}$ wrong when the key says $-\frac{\sqrt{2}}{2}$.

The correct way: They are equal — rationalizing $-\frac{1}{\sqrt{2}}$ gives $-\frac{\sqrt{2}}{2}$. Both are correct exact forms.

Mistake 3: Computing the reference angle from 90° instead of 180°

Where it slips in: Quadrant II reductions, where students subtract from $90°$ out of habit.

Don't do this: Writing the reference angle as $135° - 90° = 45°$ by coincidence-correct arithmetic that fails on other angles.

The correct way: For a Quadrant II angle, the reference angle is $180°$ minus the angle.

Key Takeaways

• Cos 135 degrees equals $-\frac{\sqrt{2}}{2}$ (about $-0.7071$), an exact value because $135°$ is a standard angle.

• The reference angle is $45°$, which gives the magnitude $\frac{\sqrt{2}}{2}$; Quadrant II makes it negative.

• In radians, $\cos 135° = \cos\left(\frac{3\pi}{4}\right)$.

• The two exact forms $-\frac{\sqrt{2}}{2}$ and $-\frac{1}{\sqrt{2}}$ are equal — and the most common mistake is keeping the sign positive.

Sharpen Your Cos 135 Degrees - Three Practice Problems

1. Evaluate $\cos 135° \cdot \sin 135°$ (use $\sin 135° = \frac{\sqrt{2}}{2}$).

2. Find the reference angle of $135°$ and rebuild $\cos 135°$ from scratch.

3. Show that $\cos 135° = -\cos 45°$ and confirm both magnitudes equal $0.7071$.

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