Cos 135 Degrees - Value −√2/2 Explained
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: cos135°=−\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) | cosθ (exact) | cosθ (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 cos135°=−\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: cos45°=\frac{\sqrt{2}}{2}. Since 135° is in Quadrant II, cosine is negative: cos135°=−cos45°=−\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}). cos135°=x-coordinate=−\frac{\sqrt{2}}{2}.
Method 3: Supplementary-angle identity
Using cos(180°−θ)=−cosθ: cos135°=cos(180°−45°)=−cos45°=−\frac{\sqrt{2}}{2}.
Examples of Cos 135 Degrees
Example 1
Evaluate 2cos135°.
2cos135°=2×(−\frac{\sqrt{2}}{2})=−\sqrt{2} ≈−1.4142.
Example 2
Find cos135° from its reference angle.
Wrong attempt: A student reports cos135°=\frac{\sqrt{2}}{2}, positive.
Correct: Reference angle =180°−135°=45°, apply Quadrant II sign: cos135°=−\frac{\sqrt{2}}{2}.
Example 3
Evaluate cos135°+cos45°.
−\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = 0.
Example 4
Verify cos2135°+sin2135°=1, given sin135°=\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 cos45°=\frac{\sqrt{2}}{2} and forgetting the Quadrant II sign.
The correct way: Cosine is negative in Quadrant II, so cos135°=−\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, cos135°=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
- Evaluate cos135°⋅sin135°.
- Find the reference angle of 135° and rebuild cos135° from scratch.
- Show that cos135°=−cos45° and confirm both magnitudes equal 0.7071.