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