Book A Free Math Class

# Cos 0 Degrees — Value of cos(0°) with the Unit Circle

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

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°.

BT

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.