# Trigonometric Table - Sin Cos Tan Values 0-90°

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

TL;DR

The trigonometric table gives the values of sine, cosine, tangent, cosecant, secant, and cotangent at the five standard angles: 0°, 30°, 45°, 60°, and 90°. The values come from two special right triangles - the 30-60-90 and the 45-45-90.

## What Is the Trigonometric Table?

The **trigonometric table** (also called the _trigonometry ratio table_ or _sin-cos-tan table_) is a reference chart of the values of the six trigonometric functions at the most commonly used angles. The five standard angles — **0°, 30°, 45°, 60°, 90°** — appear constantly in geometry, physics, engineering, and exam problems. Memorizing them saves time and clarifies reasoning.

Every value in the table comes from one of two sources:

1. **Direct geometry** — angles 0° and 90° (degenerate cases) and 45° (from the isosceles right triangle).
2. **30-60-90 triangle** — angles 30° and 60° (from the half of an equilateral triangle).

## The Complete Trigonometric Table

| Function | 0° | 30° | 45° | 60° | 90° |
| --- | --- | --- | --- | --- | --- |
| **sin θ** | 0 | 12\tfrac{1}{2}21​ | 12\tfrac{1}{\sqrt{2}}2​1​ | 32\tfrac{\sqrt{3}}{2}23​​ | 1 |
| **cos θ** | 1 | 32\tfrac{\sqrt{3}}{2}23​​ | 12\tfrac{1}{\sqrt{2}}2​1​ | 12\tfrac{1}{2}21​ | 0 |
| **tan θ** | 0 | 13\tfrac{1}{\sqrt{3}}3​1​ | 1 | 3\sqrt{3}3​ | Undefined |
| **cosec θ** | Undefined | 2 | 2\sqrt{2}2​ | 23\tfrac{2}{\sqrt{3}}3​2​ | 1 |
| **sec θ** | 1 | 23\tfrac{2}{\sqrt{3}}3​2​ | 2\sqrt{2}2​ | 2 | Undefined |
| **cot θ** | Undefined | 3\sqrt{3}3​ | 1 | 13\tfrac{1}{\sqrt{3}}3​1​ | 0 |

**Key observations:**

- sin⁡θ\sin θsinθ and cos⁡θ\cos θcosθ never exceed 1 in absolute value.
- sin⁡\sinsin runs 0→10 \to 10→1 as θθθ goes from 0° to 90°; cos⁡\coscos runs 1→01 \to 01→0 (mirror image).
- tan⁡90°\tan 90°tan90°, sec⁡90°\sec 90°sec90°, cosec⁡0°\cosec 0°cosec0°, and cot⁡0°\cot 0°cot0° are _undefined_ (division by zero in the ratio).
- sin⁡θ=cos⁡(90°−θ)\sin θ = \cos(90° - θ)sinθ=cos(90°−θ) — the **co-function identity** (which is why "co"-sine got its name).

## What Is the Trigonometric Table in Radians?

The same table, with angles expressed in radians:

| Function | 0 | π6\frac{\pi}{6}6π​ | π4\frac{\pi}{4}4π​ | π3\frac{\pi}{3}3π​ | π2\frac{\pi}{2}2π​ |
| --- | --- | --- | --- | --- | --- |
| **sin θ** | 0 | 12\tfrac{1}{2}21​ | 12\tfrac{1}{\sqrt{2}}2​1​ | 32\tfrac{\sqrt{3}}{2}23​​ | 1 |
| **cos θ** | 1 | 32\tfrac{\sqrt{3}}{2}23​​ | 12\tfrac{1}{\sqrt{2}}2​1​ | 12\tfrac{1}{2}21​ | 0 |
| **tan θ** | 0 | 13\tfrac{1}{\sqrt{3}}3​1​ | 1 | 3\sqrt{3}3​ | Undefined |
| **cosec θ** | Undefined | 2 | 2\sqrt{2}2​ | 23\tfrac{2}{\sqrt{3}}3​2​ | 1 |
| **sec θ** | 1 | 23\tfrac{2}{\sqrt{3}}3​2​ | 2\sqrt{2}2​ | 2 | Undefined |
| **cot θ** | Undefined | 3\sqrt{3}3​ | 1 | 13\tfrac{1}{\sqrt{3}}3​1​ | 0 |

## How Do You Remember the Trigonometric Table? (Memory Trick)

The cleanest memory trick uses the **square-root pattern**.

For sin values at 0°, 30°, 45°, 60°, 90°, take the sequence:

04,14,24,34,44\sqrt{\tfrac{0}{4}}, \sqrt{\tfrac{1}{4}}, \sqrt{\tfrac{2}{4}}, \sqrt{\tfrac{3}{4}}, \sqrt{\tfrac{4}{4}}40​​,41​​,42​​,43​​,44​​

which simplifies to:

0,12,12,32,10, \tfrac{1}{2}, \tfrac{1}{\sqrt{2}}, \tfrac{\sqrt{3}}{2}, 10,21​,2​1​,23​​,1

For cos values, _reverse the order_: 1,32,12,12,01, \tfrac{\sqrt{3}}{2}, \tfrac{1}{\sqrt{2}}, \tfrac{1}{2}, 01,23​​,2​1​,21​,0.

For tan values, divide sin⁡/cos⁡\sin / \cossin/cos:

tan⁡θ=sin⁡θcos⁡θ\tan θ = \frac{\sin θ}{\cos θ}tanθ=cosθsinθ​

For cosec, sec, cot, take the reciprocals: cosec⁡=1/sin⁡\cosec = 1/\sincosec=1/sin, sec⁡=1/cos⁡\sec = 1/\cossec=1/cos, cot⁡=1/tan⁡\cot = 1/\tancot=1/tan. Once you have the sin row, every other row follows.

## How Do You Derive the Trigonometric Values?

### From the 45-45-90 Triangle (Isosceles Right Triangle)

In a 45-45-90 triangle with legs of length 1, the hypotenuse is 2\sqrt{2}2​ (by the Pythagorean theorem). The two acute angles are both 45°.

sin⁡45°=12,cos⁡45°=12,tan⁡45°=1\sin 45° = \frac{1}{\sqrt{2}}, \quad \cos 45° = \frac{1}{\sqrt{2}}, \quad \tan 45° = 1sin45°=2​1​,cos45°=2​1​,tan45°=1

### From the 30-60-90 Triangle (Half of an Equilateral Triangle)

Cut an equilateral triangle of side 2 in half. The resulting right triangle has hypotenuse 2, shorter leg 1, longer leg 3\sqrt{3}3​. The angles are 30°, 60°, 90°.

sin⁡30°=12,cos⁡30°=32,tan⁡30°=13\sin 30° = \frac{1}{2}, \quad \cos 30° = \frac{\sqrt{3}}{2}, \quad \tan 30° = \frac{1}{\sqrt{3}}sin30°=21​,cos30°=23​​,tan30°=3​1​

sin⁡60°=32,cos⁡60°=12,tan⁡60°=3\sin 60° = \frac{\sqrt{3}}{2}, \quad \cos 60° = \frac{1}{2}, \quad \tan 60° = \sqrt{3}sin60°=23​​,cos60°=21​,tan60°=3​

### From the Degenerate Cases (0° and 90°)

At 0°: the opposite side has length 0, so sin⁡0°=0\sin 0° = 0sin0°=0 and cos⁡0°=1\cos 0° = 1cos0°=1.

At 90°: the adjacent side has length 0, so sin⁡90°=1\sin 90° = 1sin90°=1, cos⁡90°=0\cos 90° = 0cos90°=0, and tan⁡90°\tan 90°tan90° is undefined.

## How Do You Extend the Trigonometric Table to All 4 Quadrants?

The standard table covers angles in the **first quadrant** (0° to 90°). For angles in the other quadrants:

| Quadrant | Angle Range | Reference Angle | sin | cos | tan |
| --- | --- | --- | --- | --- | --- |
| I | 0° to 90° | θθθ | + | + | + |
| II | 90° to 180° | 180°−θ180° - θ180°−θ | + | − | − |
| III | 180° to 270° | θ−180°θ - 180°θ−180° | − | − | + |
| IV | 270° to 360° | 360°−θ360° - θ360°−θ | − | + | − |

**Memory phrase:** _"All Students Take Calculus"_ — All positive in Q1, Sine positive in Q2, Tangent positive in Q3, Cosine positive in Q4.

**Example.** Find sin⁡150°\sin 150°sin150°. 150°150°150° is in Q2. Reference angle is 180°−150°=30°180° - 150° = 30°180°−150°=30°. Sine is positive in Q2. So sin⁡150°=sin⁡30°=12\sin 150° = \sin 30° = \tfrac{1}{2}sin150°=sin30°=21​.

## Why Does the Trigonometric Table Matter? (The Real-World GROUND)

The trigonometric table isn't a school invention. It's one of the oldest reference tables in the history of science. Hipparchus of Nicaea (c. 190–c. 120 BCE) is often called the _father of trigonometry_ — he produced the first known trigonometric table to support his astronomical calculations of star positions and lunar motion.

In modern applications, the trigonometric table runs:

- **Navigation and GPS.** Every GPS satellite calculation uses trigonometric values.
- **Engineering — structural analysis.** Forces in trusses, stress in beams, electrical phase angles — all require trig values.
- **Music synthesis.** Audio waveforms are sums of sines and cosines (Fourier series).
- **Computer graphics.** 3D rotations multiply position vectors by sine/cosine matrices.
- **Surveying.** Triangulation of distances uses standard-angle trig values directly.

## A Worked Example

Find the value of sin⁡60°+cos⁡30°\sin 60° + \cos 30°sin60°+cos30°.

**The intuitive (wrong) approach.** A student remembers that "sin and cos values reverse" and guesses sin⁡60°=12\sin 60° = \tfrac{1}{2}sin60°=21​ (the cos 60° value).

**The correct method.**
sin⁡60°=32,cos⁡30°=32\sin 60° = \tfrac{\sqrt{3}}{2}, \quad \cos 30° = \tfrac{\sqrt{3}}{2}sin60°=23​​,cos30°=23​​

sin⁡60°+cos⁡30°=32+32=3\sin 60° + \cos 30° = \tfrac{\sqrt{3}}{2} + \tfrac{\sqrt{3}}{2} = \sqrt{3}sin60°+cos30°=23​​+23​​=3​

## What Are the Most Common Mistakes With the Trigonometric Table?

### **Mistake 1: Confusing sin and cos values at 30° and 60°**

**The correct way:** sin⁡30°=12\sin 30° = \tfrac{1}{2}sin30°=21​ and sin⁡60°=32\sin 60° = \tfrac{\sqrt{3}}{2}sin60°=23​​.

### **Mistake 2: Writing tan⁡90°\tan 90°tan90° as a number**

**The correct way:** tan⁡90°\tan 90°tan90° is **undefined**.

### **Mistake 3: Forgetting the quadrant sign**

**The correct way:** Apply _All Students Take Calculus_. Q1 all positive; Q2 sin positive; Q3 tan positive; Q4 cos positive.

## The Mathematicians Who Shaped the Trigonometric Table

> **Hipparchus of Nicaea (c. 190–c. 120 BCE, Greece)** — Often called the _father of trigonometry_.
> **Aryabhata (476–550 CE, India)** — In his _Aryabhatiya_ (499 CE), refined the sine table to remarkable accuracy.
> **Bhaskara II (1114–1185, India)** — In _Siddhanta-Shiromani_, extended trigonometry with approximation formulas for the sine function and tables of finer angular subdivisions.

## A Practical Next Step

Try these three before moving on to trigonometric identities.

1. Compute sin⁡30°+cos⁡60°\sin 30° + \cos 60°sin30°+cos60°.
2. Compute tan⁡45°⋅sec⁡60°\tan 45° \cdot \sec 60°tan45°⋅sec60°.
3. Find cos⁡120°\cos 120°cos120° using the quadrant rule.