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.

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.