Trigonometric Table - Sin Cos Tan Values 0-90°

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. Memorising 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

Key observations:

• $\sin θ$ and $\cos θ$ never exceed 1 in absolute value.

• $\sin$ runs $0 \to 1$ as $θ$ goes from 0° to 90°; $\cos$ runs $1 \to 0$ (mirror image).

• $\tan 90°$, $\sec 90°$, $\cosec 0°$, and $\cot 0°$ are undefined (division by zero in the ratio).

• $\sin θ = \cos(90° - θ)$ — 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 (the SI unit for angle, and the default in calculus and physics):

Degree-to-radian conversion. $\theta_\text{rad} = \theta_\text{deg} \cdot \frac{\pi}{180}$. The standard-angle radian equivalents:

In calculus, every derivative and integral identity for trig functions assumes the angle is in radians. ($\frac{d}{dx}\sin x = \cos x$ only when $x$ is in radians.) Use the radian column whenever the context is calculus, physics, or pure-math.

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:

$$\sqrt{\tfrac{0}{4}}, \sqrt{\tfrac{1}{4}}, \sqrt{\tfrac{2}{4}}, \sqrt{\tfrac{3}{4}}, \sqrt{\tfrac{4}{4}}$$

which simplifies to:

$$0, \tfrac{1}{2}, \tfrac{1}{\sqrt{2}}, \tfrac{\sqrt{3}}{2}, 1$$

For cos values, reverse the order: $1, \tfrac{\sqrt{3}}{2}, \tfrac{1}{\sqrt{2}}, \tfrac{1}{2}, 0$.

For tan values, divide $\sin / \cos$:

$$\tan θ = \frac{\sin θ}{\cos θ}$$

For cosec, sec, cot, take the reciprocals: $\cosec = 1/\sin$, $\sec = 1/\cos$, $\cot = 1/\tan$. 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 $\sqrt{2}$ (by the Pythagorean theorem). The two acute angles are both 45°.

$$\sin 45° = \frac{1}{\sqrt{2}}, \quad \cos 45° = \frac{1}{\sqrt{2}}, \quad \tan 45° = 1$$

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 $\sqrt{3}$. The angles are 30°, 60°, 90°.

$$\sin 30° = \frac{1}{2}, \quad \cos 30° = \frac{\sqrt{3}}{2}, \quad \tan 30° = \frac{1}{\sqrt{3}}$$

$$\sin 60° = \frac{\sqrt{3}}{2}, \quad \cos 60° = \frac{1}{2}, \quad \tan 60° = \sqrt{3}$$

From the Degenerate Cases (0° and 90°)

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

At 90°: the adjacent side has length 0, so $\sin 90° = 1$, $\cos 90° = 0$, and $\tan 90°$ 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:

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

Example. Find $\sin 150°$. $150°$ is in Q2. Reference angle is $180° - 150° = 30°$. Sine is positive in Q2. So $\sin 150° = \sin 30° = \tfrac{1}{2}$.

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.

The Indian mathematician Aryabhata (476–550 CE) refined the sine table to remarkable accuracy in his Aryabhatiya — values that Indian astronomers used for 1,000 years before the West rediscovered them. The modern word sine comes through a chain of translations from Aryabhata's jya.

In modern applications, the trigonometric table runs:

• Navigation and GPS. Every GPS satellite calculation uses trigonometric values. Modern receivers compute them on-chip rather than looking them up, but the underlying values are the same.

• 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). Every digital audio sample uses precomputed trig values.

• Computer graphics. 3D rotations multiply position vectors by sine/cosine matrices.

• Surveying. Triangulation of distances uses standard-angle trig values directly.

Slide rules and physical lookup tables — paper or in books — were the standard before the 1970s. Engineers designing the Apollo missions in the 1960s used trigonometric tables continuously.

A Worked Example

Find the value of $\sin 60° + \cos 30°$.

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

$$\sin 60° + \cos 30° \stackrel{?}{=} \tfrac{1}{2} + \tfrac{\sqrt{3}}{2} = \tfrac{1 + \sqrt{3}}{2}$$

Why it fails. The student confused $\sin 60°$ with $\cos 60°$. The "reverse" pattern means $\sin$ at one angle equals $\cos$ at the complementary angle (90° − that angle). So $\sin 60° = \cos 30°$, not $\cos 60°$.

The correct method.

$$\sin 60° = \tfrac{\sqrt{3}}{2}, \quad \cos 30° = \tfrac{\sqrt{3}}{2}$$

$$\sin 60° + \cos 30° = \tfrac{\sqrt{3}}{2} + \tfrac{\sqrt{3}}{2} = \sqrt{3}$$

Check. This is a special case of the co-function identity: $\sin θ = \cos(90° - θ)$. Since $\sin 60° = \cos 30°$, their sum is $2 \cos 30° = \sqrt{3}$ ✓.

At Bhanzu, our trainers teach the co-function identity early so this "reverse" pattern stops being a memory trick and becomes a fact. The memorizer who learned only "the values reverse" without the why hits this every time.

What Are the Most Common Mistakes With the Trigonometric Table?

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

Where it slips in: Recalling which has the $\sqrt{3}/2$ value.

Don't do this: Stating $\sin 30° = \tfrac{\sqrt{3}}{2}$.

The correct way: $\sin 30° = \tfrac{1}{2}$ and $\sin 60° = \tfrac{\sqrt{3}}{2}$. The smaller angle has the smaller sin value. Memory aid: sine rises from 0 to 1 as the angle increases from 0° to 90°.

Mistake 2: Writing $\tan 90°$ as a number

Where it slips in: Stating $\tan 90° = \infty$ as if it were a value.

Don't do this: $\tan 90° = \infty$.

The correct way: $\tan 90°$ is undefined, not infinity. The ratio $\sin 90° / \cos 90° = 1/0$ has no value. The function approaches $+\infty$ from one side and $-\infty$ from the other, with no defined value at 90° itself.

Mistake 3: Forgetting the quadrant sign

Where it slips in: Computing $\sin 150°$ as $-\tfrac{1}{2}$ or $\cos 240°$ as $+\tfrac{1}{2}$.

Don't do this: Forgetting that $\sin 150° = +\tfrac{1}{2}$ (Q2: sin positive) or $\cos 240° = -\tfrac{1}{2}$ (Q3: cos negative).

The correct way: Apply All Students Take Calculus. Q1 all positive; Q2 sin positive; Q3 tan positive; Q4 cos positive. Then use the reference angle to look up the magnitude.

The Mathematicians Who Shaped the Trigonometric Table

Hipparchus of Nicaea (c. 190–c. 120 BCE, Greece) — Often called the father of trigonometry. Produced the first known trig table around 130 BCE to support his astronomical calculations. Discovered the precession of the equinoxes using his table.

Aryabhata (476–550 CE, India) — In his Aryabhatiya (499 CE), refined the sine table to remarkable accuracy. Indian astronomers used his values for centuries. The modern word sine comes from a chain of translations starting with Aryabhata's jya.

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 $\sin 30° + \cos 60°$.

2. Compute $\tan 45° \cdot \sec 60°$.

3. Find $\cos 120°$ using the quadrant rule.