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

The value of cos 20 degrees is approximately $\mathbf{0.9397}$ ($0.93969262$ 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 $\cos 30° = \tfrac{\sqrt3}{2}$ on the unit circle.

Quick Reference — Cosine Near 20 Degrees

Cos 20° lands between two special angles: $\cos 0° = 1$ above and $\cos 30° = \tfrac{\sqrt3}{2}$ below.

The exact landmarks bracketing $\cos 20°$ are $\cos 0° = 1$ and $\cos 30° \approx 0.8660$ — 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 $\cos 20° \approx 0.9397$ of its slope length as horizontal run — so the horizontal coverage is now about $6$ 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 $\theta$ has coordinates $(\cos\theta, \sin\theta)$, 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.

$$\cos 20° = 0.93969262\ldots \approx 0.9397$$

In radian mode the same keystrokes give $\cos(20\ \text{rad}) \approx 0.4081$ — 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 $\cos\theta \approx 1 - \tfrac{\theta^2}{2}$ is no longer reliable here: at $20°$ ($0.349$ rad) it gives $0.9391$, off in the fourth place and drifting, so $20°$ is past the angle where that shortcut should be trusted.

What is cos 20 degrees in radians?

The angle becomes $\frac{\pi}{9}$ rad, but the cosine value is the same $\approx 0.9397$. Converting the angle's units never changes the cosine itself.

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 $0.50$.

Why it breaks. Cosine is largest near $0°$ and shrinks as the angle grows — small angle means cosine near $1$, not near $0$. Confusing this with sine (which is small near $0°$) is the slip.

Correct. $\cos 20°$ sits between $\cos 30° = 0.8660$ and $\cos 0° = 1$, so it is about $0.94$, not $0.50$.

Example 3

A cable is anchored at $20°$ to the horizontal under $500\ \text{N}$ of tension. What is the horizontal component?

Horizontal component $= 500\cos 20° = 500 \times 0.9397 = 469.8\ \text{N}$.

Example 4

Evaluate $\cos 20° + \cos 160°$.

$\cos 160° = -\cos 20°$ (supplementary angle in quadrant II), so $\cos 20° + \cos 160° = \cos 20° - \cos 20° = 0$.

Example 5

A roof slopes at $20°$. For a $9$-metre rafter, how much horizontal span does it cover?

Span $= 9\cos 20° = 9 \times 0.9397 = 8.46\ \text{m}$.

Cos 20 degrees — where students trip up

A few recurring habits cause most errors on this non-special angle.

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.

Don't do this: guessing $\cos 20° \approx 0.3$ because $20°$ "feels small."

The correct way: cosine is near $1$ for small angles and shrinks toward $0$ at $90°$ — the opposite of sine.

Mistake 2: Expecting an exact surd

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

Don't do this: trying to write $\cos 20°$ as a tidy radical.

The correct way: $20°$ is not a special angle, so $\cos 20°$ is reported as the decimal $0.9397$. (Its closed form is a cube-root expression from solving a cubic — not something to memorise.) The memoriser who knows only the special-angle table switches to a calculator here, which is the honest move.

Mistake 3: Overusing the small-angle approximation

Where it slips in: applying $\cos\theta \approx 1 - \tfrac{\theta^2}{2}$ at $20°$ and trusting all four places.

Don't do this: reporting the approximation $0.9391$ as the precise value.

The correct way: at $20°$ the approximation already drifts (it gives $0.9391$ versus the true $0.9397$). Past about $15°$, use a calculator for accuracy.

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 between $\cos 0°$ and $\cos 30°$ confirms it.

• In radians the angle is $\frac{\pi}{9}$, but the cosine value stays at $\approx 0.9397$.

• 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.

Where To Go From Here — Three Problems

1. State $\cos 20°$ to four decimal places.

2. Show that $\cos 20°$ lies between $\cos 30°$ and $\cos 0°$.

3. Find the horizontal component of a $300\ \text{N}$ force directed $20°$ above the horizontal.

If #2 confused you, recall that cosine decreases as the angle grows toward $90°$. Want a live Bhanzu trainer to walk through more cosine-value problems? Book a free demo class — online globally