# Trigonometric Chart - Values Table & How to Remember

## What Is A Trigonometric Chart?

A trigonometric chart is a table of the values of the six trigonometric ratios at the standard angles, written so you can look up any value at a glance. The standard angles are 0°, 30°, 45°, 60°, 90° — the angles that produce exact, clean values rather than the messy decimals every other angle gives.

The six ratios it lists are sine, cosine, tangent, and their reciprocals:

\[\csc\theta = \frac{1}{\sin\theta}, \qquad \sec\theta = \frac{1}{\cos\theta}, \qquad \cot\theta = \frac{1}{\tan\theta}\]

Each ratio uses the **∠** of a right triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent (the SOH-CAH-TOA rule). The chart simply records what those ratios equal at the five angles worth knowing by heart.

## The Trigonometric Chart (Degrees And Radians)

Here is the full chart. Read it across to look a value up; read the patterns below to rebuild it.

| Angle (deg) | Angle (rad) | sin⁡θ | cos⁡θ | tan⁡θ | csc⁡θ | sec⁡θ | cot⁡θ |
| --- | --- | --- | --- | --- | --- | --- | --- |
| 0° | 0 | 0 | 1 | 0 | undefined | 1 | undefined |
| 30° | \(\frac{\pi}{6}\) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{3}}\) | 2 | \(\frac{2}{\sqrt{3}}\) | \(\sqrt{3}\) |
| 45° | \(\frac{\pi}{4}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{\sqrt{2}}\) | 1 | \(\sqrt{2}\) | \(\sqrt{2}\) | 1 |
| 60° | \(\frac{\pi}{3}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\sqrt{3}\) | 2 | 2 | \(\frac{1}{\sqrt{3}}\) |
| 90° | \(\frac{\pi}{2}\) | 1 | 0 | undefined | 1 | undefined | 0 |

## How To Remember The Trigonometric Chart

**How do you memorise the trig chart fast?** Use the square-root rule for the sine row. Write the angles 0°, 30°, 45°, 60°, 90°, then under each put \(\frac{\sqrt{n}}{2}\) for n=0,1,2,3,4:

\[\sin: \quad \frac{\sqrt{0}}{2},\, \frac{\sqrt{1}}{2},\, \frac{\sqrt{2}}{2},\, \frac{\sqrt{3}}{2},\, \frac{\sqrt{4}}{2}\]

That simplifies to 0, \(\frac{1}{2}\), \(\frac{1}{\sqrt{2}}\), \(\frac{\sqrt{3}}{2}\), 1. Then:

- Cosine is the sine row _reversed_ — same five numbers, read right to left.
- Tangent is \(\frac{\sin}{\cos}\) — divide the two rows cell by cell.
- Cosecant, secant, cotangent are just the reciprocals of sine, cosine, and tangent.

Rebuild the sine row and the entire chart follows. There's no need to store thirty separate facts.

## Signs Of The Ratios Across The Quadrants

The chart covers 0° to 90° (the first **quadrant**), where every ratio is positive. Past 90°, signs change depending on the quadrant the angle lands in. The rule is **ASTC** — read anticlockwise from Quadrant I:

- Quadrant I (0°–90°): All ratios positive.
- Quadrant II (90°–180°): Sine and cosecant positive; the rest negative.
- Quadrant III (180°–270°): Tangent and cotangent positive; the rest negative.
- Quadrant IV (270°–360°): Cosine and secant positive; the rest negative.

To find a ratio outside the first quadrant, use the chart value of the **reference angle** and attach the sign ASTC gives.

## Examples Using The Trigonometric Chart

### Example 1
**Find the value of \(\sin 30° + \cos 60°\).**
From the chart, \(\sin 30° = \frac{1}{2}\) and \(\cos 60° = \frac{1}{2}\).  
\(\frac{1}{2} + \frac{1}{2} = 1\)

### Example 2
**Evaluate \(\tan 45°\).**
At 45°, the triangle is isosceles, so opposite equals adjacent, and tangent is their ratio:
\(\tan 45° = \frac{1}{1} = 1\).

### Example 3
**Find \(\sec 60°\).**
Secant is the reciprocal of cosine:
\(\cos 60° = \frac{1}{2}\)  
\(\sec 60° = \frac{1}{\cos 60°} = 2\).

### Example 6
**A ramp rises at 30° over a horizontal run of 4 m. How high is the top of the ramp?**
Height is the opposite side, so use tangent:
\(\tan 30° = \frac{h}{4} \Rightarrow h = 4 \times \tan 30° = \approx 2.31 m\).

## Why A Chart Of Exact Values Exists At All

Before calculators, every navigator, astronomer, and engineer carried trig tables — pages of values worked out by hand. The chart is a frozen calculation: the hard work done once, so it never has to be done again.

## Key Takeaways

- A **trigonometric chart** tabulates sin, cos, tan, csc, sec, cot at 0°, 30°, 45°, 60°, 90° in both degrees and radians.
- The chart gives exact values, which keeps later algebra precise where decimals would drift.