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# Cos 20 Degrees — Value of cos(20°) and How to Find It

[Trigonometry](/content/tag/trigonometry/index.html)

TL;DR

The value of cos 20 degrees is approximately 0.9397 — and 20° is not a special angle, so there is no clean exact surd for it. This article shows how to find cos(20°) honestly, gives the decimal and radian form, and places it among the standard angles.

Last updated on June 13, 2022

The value of **cos 20 degrees** is approximately 0.9397 (0.93969262 to eight places). Because 20° is not a special angle, cos(20°) has no simple exact radical — it is read off a calculator, and it sits between cos(0°) = 1 and cos(30°) = √3/2 on the unit circle.

## Quick Reference — Cosine Near 20 Degrees

Cos 20° lands between two special angles: cos(0°) = 1 above and cos(30°) = √3/2 below.

| Angle (degrees) | Angle (radians) | cos(θ) | Special angle? |
| --- | --- | --- | --- |
| 0° | 0 | 1.0000 | Yes (exact) |
| 10° | π/18 | 0.9848 | No |
| 15° | π/12 | 0.9659 | No (has a surd form) |
| **20°** | **π/9** | **0.9397** | No — decimal only |
| 30° | π/6 | 0.8660 | Yes (√3/2) |
| 45° | π/4 | 0.7071 | Yes (√2/2) |
| 60° | π/3 | 0.5000 | Yes (1/2) |

The exact landmarks bracketing cos(20°) are cos(0°) = 1 and cos(30°) ≈ 0.8660 — and 0.9397 sits comfortably between them.

## Where cos 20 Degrees Shows Up

A 20-degree angle is steep enough that cosine clearly dips below 1. A roof pitched at 20°, or a conveyor belt inclined at 20°, keeps cos(20°) ≈ 0.9397 of its slope length as horizontal run — so the horizontal coverage is now about 66% shorter than the slope, a difference engineers must account for. The value also appears in resolving forces: a 20°-angled cable carries cos(20°) of its tension horizontally, which matters in bridge and crane design.

## What Does cos 20 Degrees Mean?

On the unit circle (radius 1, centred at the origin), the point at angle θ has coordinates (cos(θ),sin(θ)), and cosine is the x-coordinate.

At 20° the radius has turned a clear amount off the positive x-axis, lifting the point to about (0.9397,0.3420) — so its x-coordinate, cos(20°), is about 0.9397.

## How Do You Find The Value of cos 20 Degrees?

There is no surd to reach for, because 20° is not a special angle. Here is how to pin the value down honestly.

### **Method 1: Calculator (set to degree mode)**

Enter cos(20) with the calculator in **DEG** mode.

cos 20° = 0.93969262… ≈ 0.9397

In radian mode the same keystrokes give cos(20 rad) ≈ 0.4081 — a different number, since 20 radians is about 1146°, several full turns. The mode is essential.

### **Method 2: Reference-angle and bracketing reasoning**

20° is already in the first quadrant, so its reference angle is itself and cosine is positive. To sanity-check the calculator value without one, bracket it between the special angles either side:

cos(30°) = 0.8660 < cos(20°) < cos(0°) = 1

Cosine decreases as the angle grows from 0° to 90°, so cos(20°) must fall between cos(30°) and cos(0°) — and 0.9397 does. The small-angle approximation cos(θ) ≈ 1−θ²/2 is no longer reliable here.

### **What is cos 20 degrees in radians?**

The angle becomes π/9 rad, but the cosine value is the same ≈ 0.9397.

## Examples Using cos 20 Degrees

### Example 1

**State cos(20°) to four decimal places.**

From a degree-mode calculator, cos(20°) = 0.9397.

### Example 2 (wrong path first)

**Without a calculator, decide whether cos(20°) is closer to 0.94 or to 0.50.**

_Wrong attempt._ A student reasons "20° is small, so cosine should be small too" and guesses 0.50.

_Correct._ cos(20°) sits between cos(30°) = 0.8660 and cos(0°) = 1, so it is about 0.94.

### Example 3

**A cable is anchored at 20° to the horizontal under 500 N of tension. What is the horizontal component?**

Horizontal component = 500 cos(20°) = 500 × 0.9397 = 469.8 N.

### Example 4

**Evaluate cos(20°) + cos(160°).**

cos(160°) = −cos(20°), so cos(20°) + cos(160°) = 0.

### Example 5

**A roof slopes at 20°. For a 999-metre rafter, how much horizontal span does it cover?**

Span = 9 cos(20°) = 9 × 0.9397 = 8.46 m.

## Cos 20 degrees — where students trip up

### Mistake 1: Thinking a small angle means a small cosine

**Where it slips in:** carrying over the intuition that "sin of a small angle is small" to cosine.

### Mistake 2: Expecting an exact surd

**Where it slips in:** assuming 20° has a clean value like 30° or 45°.

### Mistake 3: Overusing the small-angle approximation

**Where it slips in:** applying cos(θ)≈1−θ²/2 at 20° and trusting all four places.

## Key Takeaways

- **Cos 20 degrees** is approximately 0.9397 — a decimal, not a clean surd.
- 20° is not a special angle, so the value comes from a calculator; bracketing confirms it.
- In radians the angle is π/9, but the cosine value stays at ≈0.9397.
- The small-angle approximation is no longer reliable at 20° — use a calculator.
- The biggest slip is assuming a small angle gives a small cosine; cosine is near 1 for small angles.