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) | $\sin\theta$ | $\cos\theta$ | $\tan\theta$ | $\csc\theta$ | $\sec\theta$ | $\cot\theta$ |
|---|---|---|---|---|---|---|---|
$0°$ | $0$ | $0$ | $1$ | $0$ | undefined | $1$ | undefined |
$30°$ | $\frac{\pi}{6}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ | $2$ | $\frac{2}{\sqrt{3}}$ | $\sqrt{3}$ |
$45°$ | $\frac{\pi}{4}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | $1$ | $\sqrt{2}$ | $\sqrt{2}$ | $1$ |
$60°$ | $\frac{\pi}{3}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ | $\frac{2}{\sqrt{3}}$ | $2$ | $\frac{1}{\sqrt{3}}$ |
$90°$ | $\frac{\pi}{2}$ | $1$ | $0$ | undefined | $1$ | undefined | $0$ |
For the same values arranged around a circle with their coordinate meaning, see the unit circle. For the radian-first framing of the same numbers, see trigonometric ratios in radians.
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 — the quarter of the plane where both coordinates are positive), 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 (the acute angle to the nearest horizontal axis) and attach the sign ASTC gives. So $\cos 120°$ uses the chart's $\cos 60° = \frac{1}{2}$, made negative because $120°$ is in Quadrant II: $\cos 120° = -\frac{1}{2}$.
Examples Using The Trigonometric Chart
Example 1
Find the value of $\sin 30° + \cos 60°$.
From the chart, $\sin 30° = \frac{1}{2}$ and $\cos 60° = \frac{1}{2}$.
$$\frac{1}{2} + \frac{1}{2} = 1$$
Final answer: $1$.
Example 2
Evaluate $\tan 45°$ — and watch a common slip. A frequent first instinct is to assume $\tan 45°$ must be a "nice" fraction like $\frac{1}{2}$, since $45°$ sits halfway up the chart.
That guess comes from reading the position of the row instead of the value. Halfway up the table does not mean halfway between $0$ and $1$. Check the definition: at $45°$ the triangle is isosceles, so opposite equals adjacent, and tangent is their ratio.
$$\tan 45° = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{1} = 1$$
The chart confirms $\tan 45° = 1$, not $\frac{1}{2}$.
Final answer: $1$.
Example 3
Find $\sec 60°$.
Secant is the reciprocal of cosine.
$$\cos 60° = \frac{1}{2}$$
$$\sec 60° = \frac{1}{\cos 60°} = \frac{1}{\frac{1}{2}} = 2$$
Final answer: $\sec 60° = 2$.
Example 4
Evaluate $\sin^2 30° + \cos^2 30°$.
$$\sin 30° = \frac{1}{2} \Rightarrow \sin^2 30° = \frac{1}{4}$$
$$\cos 30° = \frac{\sqrt{3}}{2} \Rightarrow \cos^2 30° = \frac{3}{4}$$
$$\frac{1}{4} + \frac{3}{4} = 1$$
Final answer: $1$. This is the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ — true for every angle, not just $30°$.
Example 5
Use the chart and ASTC to find $\cos 150°$.
$150°$ is in Quadrant II, where cosine is negative. Its reference angle to the horizontal axis is $180° - 150° = 30°$.
$$\cos 150° = -\cos 30° = -\frac{\sqrt{3}}{2}$$
Final answer: $-\frac{\sqrt{3}}{2}$.
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, run is the adjacent side, so use tangent.
$$\tan 30° = \frac{h}{4}$$
$$h = 4 \times \tan 30° = 4 \times \frac{1}{\sqrt{3}} = \frac{4}{\sqrt{3}} \approx 2.31 \text{ m}$$
Final answer: about $2.31$ m. The chart turned a real measurement into an answer in one line.
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 so a sailor at sea could fix a position without recomputing a sine from scratch. The chart is a frozen calculation: the hard work done once, so it never has to be done again.
Two reasons it still earns its place:
Exactness. A calculator gives $\cos 30° = 0.8660254...$, but the chart gives $\frac{\sqrt{3}}{2}$ — the exact value, which keeps later algebra clean and lets surds cancel instead of accumulating rounding error.
Speed and insight. Knowing the five standard angles cold means recognising structure: spotting that $\sin 30° = \cos 60°$ reveals the cofunction relationship between complementary angles, something a decimal hides completely.
That is where this is heading: the chart is the launchpad for identities, the unit circle, and eventually the wave functions of calculus.
Tripping Points To Avoid
Mistake 1: Swapping the tangent of $30°$ and $60°$
Where it slips in: Recalling the tangent row under time pressure.
Don't do this: Write $\tan 30° = \sqrt{3}$ and $\tan 60° = \frac{1}{\sqrt{3}}$.
The correct way: Tangent increases with the angle from $0°$ to $90°$, so the bigger angle has the bigger tangent: $\tan 30° = \frac{1}{\sqrt{3}}$ (small), $\tan 60° = \sqrt{3}$ (large). Rebuilding from $\frac{\sin}{\cos}$ settles it every time — this exact swap is the most common chart error students make.
Mistake 2: Forgetting which values are undefined
Where it slips in: Reciprocal ratios at $0°$ and $90°$.
Don't do this: Write $\tan 90° = \infty$ as if it were a number, or treat $\csc 0°$ as $0$.
The correct way: $\tan 90°$, $\sec 90°$, $\csc 0°$, and $\cot 0°$ are undefined — they come from dividing by zero. Write "undefined", not a value.
Mistake 3: Using the chart value without applying the quadrant sign
Where it slips in: Angles beyond $90°$, like $\cos 120°$ or $\sin 210°$.
Don't do this: Read $\cos 120°$ as $+\frac{1}{2}$ straight from the $60°$ row.
The correct way: Take the chart value of the reference angle, then attach the ASTC sign. The reference-angle step feels skippable for $120°$, but skipping it is exactly where the sign goes wrong.
Key Takeaways
A trigonometric chart tabulates sin, cos, tan, csc, sec, cot at $0°, 30°, 45°, 60°, 90°$ in both degrees and radians.
Rebuild the sine row as $\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$; cosine is that row reversed; the rest follow.
$\tan 90°$, $\sec 90°$, $\csc 0°$, $\cot 0°$ are undefined (division by zero).
Beyond $90°$, use the reference-angle value plus the ASTC quadrant sign.
The chart gives exact surd values, which keeps later algebra precise where decimals would drift.
To master the trigonometric chart with a teacher walking you through each row, explore Bhanzu's trigonometry tutor, a high school math tutor, or math classes online.
Read More
Trigonometric Table — an extended reference table of ratio values.
Trigonometry Formulas — the full formula set that builds on the chart values.
Reference Angle — finding the acute angle that unlocks values past 90°.
Sin Cos Tan — the three primary ratios in detail.
Trigonometric Identities — the identities that build on the chart values.
Reciprocal Identities — how csc, sec, and cot relate to sin, cos, and tan.
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