Trigonometric Table - Sin Cos Tan Values 0-90°
Trigonometric Table - Sin Cos Tan Values 0-90°
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:
- Direct geometry — angles 0° and 90° (degenerate cases) and 45° (from the isosceles right triangle).
- 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}}21 | 32\tfrac{\sqrt{3}}{2}23 | 1 |
| cos θ | 1 | 32\tfrac{\sqrt{3}}{2}23 | 12\tfrac{1}{\sqrt{2}}21 | 12\tfrac{1}{2}21 | 0 |
| tan θ | 0 | 13\tfrac{1}{\sqrt{3}}31 | 1 | 3\sqrt{3}3 | Undefined |
| cosec θ | Undefined | 2 | 2\sqrt{2}2 | 23\tfrac{2}{\sqrt{3}}32 | 1 |
| sec θ | 1 | 23\tfrac{2}{\sqrt{3}}32 | 2\sqrt{2}2 | 2 | Undefined |
| cot θ | Undefined | 3\sqrt{3}3 | 1 | 13\tfrac{1}{\sqrt{3}}31 | 0 |
Key observations:
- sinθ\sin θsinθ and cosθ\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).
- tan90°\tan 90°tan90°, sec90°\sec 90°sec90°, cosec0°\cosec 0°cosec0°, and cot0°\cot 0°cot0° are undefined (division by zero in the ratio).
- sinθ=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}}21 | 32\tfrac{\sqrt{3}}{2}23 | 1 |
| cos θ | 1 | 32\tfrac{\sqrt{3}}{2}23 | 12\tfrac{1}{\sqrt{2}}21 | 12\tfrac{1}{2}21 | 0 |
| tan θ | 0 | 13\tfrac{1}{\sqrt{3}}31 | 1 | 3\sqrt{3}3 | Undefined |
| cosec θ | Undefined | 2 | 2\sqrt{2}2 | 23\tfrac{2}{\sqrt{3}}32 | 1 |
| sec θ | 1 | 23\tfrac{2}{\sqrt{3}}32 | 2\sqrt{2}2 | 2 | Undefined |
| cot θ | Undefined | 3\sqrt{3}3 | 1 | 13\tfrac{1}{\sqrt{3}}31 | 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}, 10,21,21,23,1
For cos values, reverse the order: 1,32,12,12,01, \tfrac{\sqrt{3}}{2}, \tfrac{1}{\sqrt{2}}, \tfrac{1}{2}, 01,23,21,21,0.
For tan values, divide sin/cos\sin / \cossin/cos:
tanθ=sinθcosθ\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°.
sin45°=12,cos45°=12,tan45°=1\sin 45° = \frac{1}{\sqrt{2}}, \quad \cos 45° = \frac{1}{\sqrt{2}}, \quad \tan 45° = 1sin45°=21,cos45°=21,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°.
sin30°=12,cos30°=32,tan30°=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°=31
sin60°=32,cos60°=12,tan60°=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 sin0°=0\sin 0° = 0sin0°=0 and cos0°=1\cos 0° = 1cos0°=1.
At 90°: the adjacent side has length 0, so sin90°=1\sin 90° = 1sin90°=1, cos90°=0\cos 90° = 0cos90°=0, and tan90°\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 sin150°\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 sin150°=sin30°=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 sin60°+cos30°\sin 60° + \cos 30°sin60°+cos30°.
The intuitive (wrong) approach. A student remembers that "sin and cos values reverse" and guesses sin60°=12\sin 60° = \tfrac{1}{2}sin60°=21 (the cos 60° value).
The correct method. sin60°=32,cos30°=32\sin 60° = \tfrac{\sqrt{3}}{2}, \quad \cos 30° = \tfrac{\sqrt{3}}{2}sin60°=23,cos30°=23
sin60°+cos30°=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: sin30°=12\sin 30° = \tfrac{1}{2}sin30°=21 and sin60°=32\sin 60° = \tfrac{\sqrt{3}}{2}sin60°=23.
Mistake 2: Writing tan90°\tan 90°tan90° as a number
The correct way: tan90°\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.
- Compute sin30°+cos60°\sin 30° + \cos 60°sin30°+cos60°.
- Compute tan45°⋅sec60°\tan 45° \cdot \sec 60°tan45°⋅sec60°.
- Find cos120°\cos 120°cos120° using the quadrant rule.