Trigonometric Table (Trig Table)

Why the table only needs 0° to 45°

The cofunction identities say sin θ = cos(90° − θ), tan θ = cot(90° − θ) and sec θ = csc(90° − θ). So the sine column from 0° to 45° is the cosine column read from the bottom up: sin 25° = 0.4226 and cos 65° = 0.4226. Older printed tables used this to halve their size, with the angle running down on the left for 0° to 45° and up on the right for 45° to 90°. Selecting a row in the table highlights its complementary angle in blue.

Angles beyond 90°: reference angles and signs

• For any angle from 0° to 360°, find its reference angle, the acute angle between the terminal side and the x-axis. In quadrant II it is 180° − θ, in quadrant III it is θ − 180°, and in quadrant IV it is 360° − θ. Look up the reference angle in the table, then apply the sign of the quadrant.
• The signs follow the mnemonic All Students Take Calculus: all six functions are positive in quadrant I, only sine (and cosecant) in II, only tangent (and cotangent) in III, and only cosine (and secant) in IV. For example sin 150° has reference angle 30° and sine is positive in quadrant II, so sin 150° = 0.5000. Cosine is negative there, so cos 150° = −0.8660. In quadrant III tangent is positive, so tan 225° = tan 45° = 1.0000.

Degrees and radians

A full turn is 360° or 2π radians, so 180° = π and 1° = π/180 radians, about 0.0175. To convert degrees to radians multiply by π/180; to go back multiply by 180/π. The table shows both the exact multiple of π (for example 30° = π/6) and the decimal. The calculator accepts either unit, and understands π, pi and the degree sign, so π/6, 30° and 30 all give the same row.

Why tan 90° is undefined

Tangent is sine divided by cosine, and cos 90° = 0, so tan 90° would need a division by zero. Near 90° the value explodes: tan 89° = 57.2900, and it keeps growing as the angle gets closer, which is why the graph has a vertical asymptote there. The same happens for tan 270° and sec at 90° and 270°, and for cot and csc at 0°, 180° and 360°. The table writes undefined in those cells instead of a number.

Exact values for special angles

At 0°, 30°, 45°, 60° and 90° the functions have exact values with fractions and square roots: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2 and tan 30° = √3/3. Exams usually want these exact forms, not decimals. The Exact values tab lists them for every special angle in all four quadrants, together with 15° and 75°, and the unit circle shows each as an (cos θ, sin θ) point.