Cos 0 Degrees — Value of cos(0°) with the Unit Circle
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Cos 0 Degrees — Value of cos(0°) with the Unit Circle
TL;DR
The value of cos 0 degrees is exactly 1 (1.0000 as a decimal). This article shows why with both the unit circle and the right triangle, gives a full standard-angle cosine table in degrees and radians, and clears up the mistakes that trip students on cos 0°.
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Last updated on June 13, 2026 5 min read
Quick Answer:
- Result: cos 0° = 1
- In radians: cos 0 = 1 (since 0° = 0 rad)
- Notation: exact integer — 1, or 1.0000 as a decimal
- Method shown: unit circle (x-coordinate at 0°) and right-triangle ratio (adjacent ÷ hypotenuse)
- Exact form: 1 — no rounding, no surd; this is one of the clean special-angle values
Quick Reference — Cosine of The Standard Angles
These are the special angles worth memorising. Cos 0° sits at the top, where cosine is at its largest.
| Angle (degrees) | Angle (radians) | cos θ | Decimal |
|---|---|---|---|
| 0° | 0 | 1 | 1.0000 |
| 30° | π/6 | √3/2 | 0.8660 |
| 45° | π/4 | √2/2 | 0.7071 |
| 60° | π/3 | 1/2 | 0.5000 |
| 90° | π/2 | 0 | 0.0000 |
| 120° | 2π/3 | -1/2 | -0.5000 |
| 180° | π | -1 | -1.0000 |
Cosine starts at 1, falls to 0 at 90°, and reaches -1 at 180°. The value at 0° is the maximum cosine ever takes.
Where cos 0 Degrees Shows Up
Cos 0° is the value behind anything pointing straight along its reference axis. The horizontal range of a projectile launched at 0° uses cos 0° = 1, meaning all of its speed is horizontal and none is vertical. It also appears as the starting amplitude of any cosine wave — A cos(0) = A — which is why a cosine signal begins at its peak, the moment used to set the phase reference for AC voltage and oscillating systems.
What Does cos 0 Degrees Mean?
Cosine measures how much of a direction points along the horizontal. For an angle θ on the unit circle (radius 1, centred at the origin), the point at that angle has coordinates (cos θ, sin θ).
At 0°, the radius hasn't turned at all — it lies flat along the positive x-axis, ending at (1,0). Cosine reads the x-coordinate, so cos 0° = 1.
How Do You Find The Value of cos 0 Degrees?
Two methods give the same answer, and seeing both stops sin/cos from feeling like two unrelated ideas.
Method 1: Unit circle
The angle 0° (0 radians) places the point at (1,0).
cos 0° = x-coordinate = 1
Method 2: Right triangle (adjacent over hypotenuse)
Cosine of an angle in a right triangle is adjacent/hypotenuse. As the angle shrinks toward 0°, the adjacent side stretches until it equals the hypotenuse.
cos 0° = adjacent/hypotenuse = 1/1 = 1.
Is cos 0 the same in radians? Yes. 0° converts to 0 radians, and the angle is the same physical angle, so cos 0 = 1 either way. The unit only changes the label, not the point on the circle.
Examples Using cos 0 Degrees
Example 1
Evaluate cos 0° + sin 90°.
cos 0° = 1 and sin 90° = 1, so the sum is 1 + 1 = 2.
Example 2 (wrong path first)
Evaluate 5 cos 0°.
Wrong attempt. A student writes 5 cos 0° = 0, reasoning that "anything with 0 in it is 0."
Why it breaks. The 0 is the angle, not a factor being multiplied. cos 0° is a single value — and that value is 1, not 0.
Correct. 5 cos 0° = 5 × 1 = 5.
Example 3
Find cos 0° × cos 90°.
cos 0° = 1 and cos 90° = 0, so 1 × 0 = 0. The product is zero because of the 90° term, not the 0° one.
Example 4
A projectile is launched at 0° above the horizontal with speed 20 m/s. What is its horizontal speed component?
Horizontal component = 20 cos 0° = 20 × 1 = 20 m/s. All the speed is horizontal.
Example 5
Evaluate cos 0°/sin 90° + cos 180°.
1 + (-1) = 1 - 1 = 0.
Cos 0 degrees — Where Students Trip Up
A handful of slips show up again and again on this value.
Mistake 1: Reading cos 0° as 0
Where it slips in: scanning a problem fast and pattern-matching "0" to "answer is 0."
Don't do this: writing cos 0° = 0. That value belongs to sin 0° and to cos 90° — not cos 0°.
The correct way: cos 0° = 1; sin 0° = 0.
Mistake 2: Thinking the angle unit changes the value
Where it slips in: switching between degrees and radians mid-problem.
Don't do this: assuming cos 0° and cos 0 (radians) could differ.
The correct way: 0° = 0 radians, so both equal 1. The memoriser who only learned "cos 0 = 1 in radians" still gets the degree version right here — the angle is the same.
Mistake 3: Confusing cos 0° with the inverse, cos⁻¹(0)
Where it slips in: the notation cos⁻¹ looks close to cos 0.
Don't do this: writing cos⁻¹(0) = 1.
The correct way: cos 0° = 1, but cos⁻¹(0) = 90° — the inverse asks "which angle has cosine 0?" Different question entirely.
Key takeaways
- Cos 0 degrees equals 1 exactly — the maximum value cosine takes.
- On the unit circle, 0° is the point (1,0), and cosine reads the x-coordinate.
- The right-triangle view agrees: as the angle shrinks to 0°, adjacent equals hypotenuse, so the ratio is 1.
- The most common mistake is confusing cos 0°=1 with sin 0°=0 — the table fixes this.
- The value is identical in degrees and radians, since 0°=0 rad.