# Cos 135 Degrees - Value −√2/2 Explained

TL;DR  
The value of cos 135 degrees is exactly −\frac{\sqrt{2}}{2}, about −0.7071. This article shows why the value is negative (135° sits in Quadrant II), how the reference angle of 45° supplies the magnitude, a standard-angle table in degrees and radians, plus worked examples and the common mistakes.

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

### **Quick Answer:**
- **Result:** cos⁡135°=−\frac{\sqrt{2}}{2}
- **Decimal:** ≈−0.7071
- **In radians:** cos⁡(3π/4)=−\frac{\sqrt{2}}{2} 
- **Exact form:** −\frac{\sqrt{2}}{2}, equivalently −\frac{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

| Angle (degrees) | Angle (radians) | cos⁡θ (exact) | cos⁡θ (decimal) |
| --- | --- | --- | --- |
| 0° | 0 | 1 | 1.0000 |
| 30° | \dfrac{\pi}{6} | \dfrac{\sqrt{3}}{2} | 0.8660 |
| 45° | \dfrac{\pi}{4} | \dfrac{\sqrt{2}}{2} | 0.7071 |
| 60° | \dfrac{\pi}{3} | \dfrac{1}{2} | 0.5000 |
| 90° | \dfrac{\pi}{2} | 0 | 0.0000 |
| 120° | \dfrac{2\pi}{3} | −\dfrac{1}{2} | −0.5000 |
| 135° | \dfrac{3\pi}{4} | −\dfrac{\sqrt{2}}{2} | −0.7071 |
| 150° | \dfrac{5\pi}{6} | −\dfrac{\sqrt{3}}{2} | −0.8660 |
| 180° | \pi | −1 | −1.0000 |

Cos 135° and cos 45° carry the same magnitude, \dfrac{\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. In physics and robotics, a movement vector at 135° has equal-magnitude components (−\frac{\sqrt{2}}{2} across, +\frac{\sqrt{2}}{2} up). The same value appears in the trigonometric ratios behind a 45° roof rafter and in alternating-current phase math where signals sit a quarter-turn-plus apart.

## 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 at (−\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}), whose negative x-coordinate gives cos⁡135°=−\frac{\sqrt{2}}{2}.

## 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 (−\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}). cos⁡135°=x-coordinate=−\frac{\sqrt{2}}{2}.

### **Method 3: Supplementary-angle identity**
Using cos⁡(180°−θ)=−cos⁡θ: cos⁡135°=cos⁡(180°−45°)=−cos⁡45°=−\frac{\sqrt{2}}{2}.

## Examples of Cos 135 Degrees
### Example 1
**Evaluate 2cos⁡135°**.  
2cos⁡135°=2×(−\frac{\sqrt{2}}{2})=−\sqrt{2} ≈−1.4142.

### Example 2
**Find cos⁡135° from its reference angle.**  
Wrong attempt: A student reports cos⁡135°=\frac{\sqrt{2}}{2}, positive.  
Correct: Reference angle =180°−135°=45°, apply 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.

### Example 4
**Verify cos⁡2135°+sin⁡2135°=1, given sin⁡135°=\frac{\sqrt{2}}{2}.**  
(−\frac{\sqrt{2}}{2})² + (\frac{\sqrt{2}}{2})² = \frac{2}{4} + \frac{2}{4} = 1.

### Example 5
**Express 135° in radians and evaluate cos⁡(3π/4).**  
135°=\frac{3\pi}{4}, so cos⁡(3π/4)=−\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.  
**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.  
**The correct way:** They are equal — rationalizing −\frac{1}{\sqrt{2}} gives −\frac{\sqrt{2}}{2}.

### Mistake 3: Computing the reference angle from 90° instead of 180°
**Where it slips in:** Students subtract from 90° instead of using 180° for Quadrant II 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°, giving magnitude \frac{\sqrt{2}}{2}; Quadrant II makes it negative.
- In radians, cos⁡135°=cos⁡(3π/4).
- The two exact forms −\frac{\sqrt{2}}{2} and −\frac{1}{\sqrt{2}} are equal — common mistake is keeping the sign positive.

## Sharpen Your Cos 135 Degrees - Three Practice Problems
1. Evaluate cos⁡135°⋅sin⁡135°.
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.