Trigonometry begins with a right triangle and a question: given one acute angle and one side, what are the rest? The answer is a set of ratios — sine, cosine, tangent — that depend only on the angle, not on the triangle’s size, because all right triangles sharing an angle are similar and their corresponding sides are in the same proportion. That size-independence is the whole subject in miniature: the ratios are functions of the angle, and the functions extend beyond the right triangle, beyond 90°, around the full circle, and into every periodic phenomenon in science. This guide defines the ratios, extends them, and applies them, keeping the triangle and the circle in view together.

The ratios: sine, cosine, tangent, and their reciprocals

For an acute angle θ in a right triangle, with the hypotenuse as the longest side: sin θ is the opposite side divided by the hypotenuse, cos θ is the adjacent side divided by the hypotenuse, and tan θ is the opposite divided by the adjacent — equal to sin θ / cos θ. In the 3-4-5 triangle, the angle opposite the side of 3 has sin = 3/5 = 0.6, cos = 4/5 = 0.8, tan = 3/4 = 0.75. The reciprocal ratios complete the set of six: cosecant (csc θ = 1/sin θ = 5/3), secant (sec θ = 1/cos θ = 5/4), and cotangent (cot θ = 1/tan θ = 4/3). The mnemonics (SOH-CAH-TOA) encode the first three; the reciprocals are their inverses, and no separate memory is needed for them.

The ratios are fixed by the angle because the triangles are similar: double every side of the 3-4-5 triangle and the angle is unchanged, the sides are 6-8-10, and sin is still 6/10 = 0.6. That is why a table of sines is possible at all, and why the ratios are functions — each angle has exactly one sine, one cosine, one tangent. The special angles are worth knowing exactly: sin 30° = 0.5, cos 30° = √3/2 ≈ 0.866, tan 30° = 1/√3 ≈ 0.577; sin 45° = cos 45° = 1/√2 ≈ 0.707, tan 45° = 1; sin 60° = √3/2, cos 60° = 0.5, tan 60° = √3 ≈ 1.732. These follow from the geometry of the equilateral triangle halved and the square halved diagonally, and they are the values against which calculator output should be sanity-checked.

The unit circle: every angle, and why the functions repeat

Place the angle at the origin, its vertex at the centre of a circle of radius 1, and its terminal side meeting the circle at a point: cos θ is that point’s x-coordinate and sin θ its y-coordinate. For acute angles this reproduces the triangle ratios exactly; for angles beyond 90° it extends them, with signs determined by the quadrant — sine positive in the first and second quadrants, cosine positive in the first and fourth, tangent positive in the first and third (the CAST rule, read from the fourth quadrant). The point’s distance from the origin is 1, so x² + y² = 1 for every angle, which is the Pythagorean identity sin²θ + cos²θ = 1 — not an empirical pattern but a restatement of the circle’s definition, and the identity from which the other two Pythagorean forms follow by division: 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ.

Going around the circle returns to the start, so the functions are periodic: sin(θ + 360°) = sin θ, and the same for cosine, with tangent repeating every 180°. Negative angles rotate clockwise, and sin(−θ) = −sin θ while cos(−θ) = cos θ — sine is odd, cosine even, a symmetry that halves the work in many problems. The reference angle — the acute angle between the terminal side and the x-axis — gives the magnitude, and the quadrant gives the sign: every trigonometric evaluation is a reference-angle computation plus a sign decision, and separating the two is how sign errors are avoided.

The identities: angle sums, double angles, and solving equations

The angle-sum formulas express the sine and cosine of a sum in terms of the parts: sin(A + B) = sin A cos B + cos A sin B, and cos(A + B) = cos A cos B − sin A sin B. Setting B = A gives the double-angle formulas: sin 2A = 2 sin A cos A, and cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A. The sum-to-product and product-to-sum formulas run the same relationships in the other direction, and together the identities are a complete toolkit: any trigonometric expression can be rewritten, combined, or reduced until it is solvable, and the skill is recognising which form the problem needs rather than memorising the list.

Solving a trigonometric equation means finding every angle that satisfies it, not one: sin θ = 0.5 is satisfied by 30° and by 150°, and by every coterminal angle of both, and a solution that reports only 30° is incomplete. The inverse functions (arcsin, arccos, arctan) return one principal value each — arcsin returns values between −90° and 90° — and the full solution set is built from the principal value, the reference angle, the quadrant signs, and the periodicity. State the interval when the problem gives one (0° to 360°, or 0 to 2π), and list every solution inside it.

Solving triangles and measuring heights

The ratios solve the triangle from any sufficient data: a right triangle with one acute angle and the hypotenuse gives the legs directly (opposite = hypotenuse × sin, adjacent = hypotenuse × cos), and a right triangle with two sides gives the angles through the inverse ratios. The angle of elevation and the angle of depression are the same angle measured from the horizontal — elevation looking up, depression looking down — and the height of an inaccessible object follows from the angle and the distance: a tree 30 m away subtending an angle of elevation of 35° is 30 × tan 35° = 21.0 m tall, plus the observer’s eye height. Two observations from two distances give the height without knowing either distance, by solving the two equations together — the method of surveying, triangulation, and every height-from-a-photograph problem.

For triangles without a right angle, the triangle solution on this site uses the sine and cosine rules, which are trigonometry’s extension to the general triangle, and the trigonometry calculation carries the ratio and identity arithmetic. The two calculators divide the subject at its natural joint: ratios and functions here, whole triangles there.

Radians: the natural unit, and the conversion that must not be skipped

Angles are measured in degrees (360 to a circle, a Babylonian inheritance) and in radians (2π to a circle, the ratio of arc length to radius). The radian is the natural unit for calculus — the derivative of sin x is cos x only when x is in radians — and the conversion is 180° = π radians, so degrees × π/180 = radians and radians × 180/π = degrees. A 30° angle is π/6 radians (0.5236), 45° is π/4 (0.7854), 90° is π/2 (1.5708). Calculators have a mode switch, and the mode is the commonest source of wrong answers in trigonometry: sin(30) is 0.5 in degree mode and −0.988 in radian mode, and the calculator reports both without comment. Check the mode before trusting the output, and check the output against a special angle after it.

Periodic motion: where the functions live in the world

Every periodic phenomenon is described, exactly or approximately, by sines and cosines: sound waves, light waves, alternating current, tides, pendulums, and the seasonal cycle. A sinusoidal quantity is A·sin(ωt + φ), where A is the amplitude, ω the angular frequency (2π times the frequency), and φ the phase — and any periodic waveform, however complex, is a sum of sinusoids at harmonic frequencies, which is Fourier’s theorem and the basis of signal processing, audio compression, and medical imaging. The unit circle is not an abstraction imposed on the world; it is the shape the world’s oscillations trace, and the trigonometric functions are the coordinates of that shape as it turns.

Frequently asked questions

What is the difference between sine, cosine, and tangent?

They are three ratios of a right triangle’s sides for a given angle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent (equal to sine divided by cosine). The three reciprocal ratios are cosecant (1/sine), secant (1/cosine) and cotangent (1/tangent). The ratios depend only on the angle, because all right triangles sharing an angle are similar.

What is the unit circle?

A circle of radius 1 centred at the origin, used to define sine and cosine for every angle: cos θ is the x-coordinate and sin θ the y-coordinate of the point at angle θ on the circle. It extends the ratios beyond 90°, explains the signs in each quadrant, and makes the identity sin²θ + cos²θ = 1 a restatement of the circle’s equation, x² + y² = 1.

How do you convert degrees to radians?

Multiply by π/180: degrees × π/180 = radians. Thirty degrees is π/6 radians (0.5236), 45° is π/4 (0.7854), 90° is π/2 (1.5708), and 180° is π. Radians are the natural unit for calculus; check the calculator’s mode before trusting output, since sin(30) is 0.5 in degree mode and −0.988 in radian mode.

What is the Pythagorean identity?

Sin²θ + cos²θ = 1, for every angle. It follows from the unit circle: the point (cos θ, sin θ) lies on a circle of radius 1, so its coordinates satisfy x² + y² = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ.

How do you find a height using trigonometry?

Measure the angle of elevation to the top and the horizontal distance to the object, then multiply: height = distance × tan(angle), plus the observer’s eye height. A tree 30 m away at an angle of elevation of 35° is 30 × tan 35° = 21.0 m above eye level. Two observations from two distances give the height without knowing either distance, by solving the two equations together.

Why do trigonometric equations have more than one solution?

Because the functions are periodic and symmetric: sin θ = 0.5 is satisfied by 30° and 150°, and by every coterminal angle of both. The inverse functions return one principal value each, and the full solution set is built from it using the reference angle, the quadrant signs, and the periodicity. Always state the interval and list every solution inside it.