Trigonify reference
All 45 identities and formulas across 16 topics, each with its KaTeX-rendered notation and its own reference page. Every page here is free to read - no account needed. For the method behind the formulas, read the trigonometry guides.
Basics · 4
SOHCAHTOA is the mnemonic for the three ratios in a right triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. Everything else in trigonometry is built on these three. Use it whenever a problem gives you a right angle and any two of a side, an angle, or the hypotenuse.
One degree is π/180 radians, and one radian is 180/π degrees - so multiply by π/180 going into radians and by 180/π coming out. Radians are the unit calculus assumes, which is why arc length, sector area and every derivative of a trig function require them. Getting caught in degree mode is the single most common source of wrong answers on a calculator.
The arc cut off by a central angle θ on a circle of radius r has length s = rθ. θ must be in radians - the formula is only this clean because a radian is defined as the angle whose arc equals the radius. In degrees you would need s = (πrθ)/180 instead.
A sector with central angle θ on a circle of radius r has area ½r²θ, with θ in radians. It is the whole circle's πr² scaled by the fraction θ/2π that the sector occupies. Pair it with the arc-length formula for any "slice of a circle" question.
Pythagorean · 3
The Pythagorean identity says sin²θ + cos²θ = 1 for every angle θ - it is the Pythagorean theorem read off the unit circle, where sine and cosine are the legs and the radius is 1. It is the most-used identity in the subject: rearranged it converts any sine into a cosine (and back), which is how most simplification and integration problems get started.
Dividing sin²θ + cos²θ = 1 through by cos²θ gives 1 + tan²θ = sec²θ. Reach for it whenever an expression mixes tangents and secants - it is the substitution behind the trigonometric-substitution integral for √(x² + a²).
Dividing sin²θ + cos²θ = 1 through by sin²θ gives 1 + cot²θ = csc²θ. It is the cotangent-cosecant counterpart of the tangent-secant form, and the one to use when an expression is written in terms of sines rather than cosines.
Reciprocal · 3
Secant is defined as the reciprocal of cosine, so sec θ = 1/cos θ. It is undefined wherever cos θ = 0 - at π/2, 3π/2 and every odd multiple of π/2 - which is exactly where the secant graph has its vertical asymptotes.
Cosecant is the reciprocal of sine, so csc θ = 1/sin θ. It is undefined wherever sin θ = 0 - at 0, π and every multiple of π. Note the mismatch students trip on: cosecant goes with sine, not cosine.
Cotangent is the reciprocal of tangent, so cot θ = 1/tan θ. The reciprocal form and the quotient form cot θ = cos θ / sin θ agree everywhere both are defined, but the quotient form is the safer one to use at angles where tan θ is zero or undefined.
Quotient · 2
Tangent is the quotient of sine and cosine: tan θ = sin θ / cos θ. This is the identity that turns a mixed expression into one written purely in sines and cosines, which is almost always the first move when simplifying or proving an identity.
Cotangent is cosine over sine: cot θ = cos θ / sin θ. Together with tan θ = sin θ / cos θ it lets you rewrite any expression in sines and cosines alone - the standard opening move for an identity proof.
Even/Odd · 3
Sine is an odd function, so flipping the sign of the angle flips the sign of the value: sin(−θ) = −sin θ. Its graph has rotational symmetry about the origin. This is what lets you fold negative angles out of an expression before applying any other identity.
Cosine is an even function: cos(−θ) = cos θ, so the sign of the angle makes no difference. Its graph is symmetric about the vertical axis. Cosine being even while sine is odd is precisely why the sum and difference formulas have the ± / ∓ pairings they do.
Tangent is odd, so tan(−θ) = −tan θ. It follows directly from sine being odd and cosine being even: the sign in the numerator flips and the denominator does not.
Cofunction · 3
The sine of an angle equals the cosine of its complement: sin(π/2 − θ) = cos θ. In a right triangle the two acute angles are complements, and one angle's opposite side is the other's adjacent side - which is where the "co" in cosine comes from.
The cosine of an angle equals the sine of its complement: cos(π/2 − θ) = sin θ. It is the mirror of the sine cofunction identity, and the reason the sine and cosine graphs are the same curve shifted by π/2.
Tangent and cotangent are cofunctions: tan(π/2 − θ) = cot θ. The same complementary-angle relationship holds for every function-and-cofunction pair - sine/cosine, tangent/cotangent, secant/cosecant.
Law of Sines · 2
In any triangle, each side divided by the sine of its opposite angle gives the same value: a/sin A = b/sin B = c/sin C. Use it for AAS, ASA, and SSA. SSA is the ambiguous case - it can yield two triangles, one, or none, so always check whether a second angle also fits.
The common ratio in the Law of Sines is not arbitrary: it equals 2R, twice the radius of the triangle's circumscribed circle. That turns the Law of Sines into a way of finding the circumradius from any one side and its opposite angle.
Law of Cosines · 2
a² = b² + c² − 2bc cos A finds the third side from two sides and the angle between them (SAS). It is the Pythagorean theorem with a correction term: when A = 90°, cos A = 0 and it collapses to a² = b² + c².
Rearranged for the angle, cos A = (b² + c² − a²)/(2bc) finds any angle from all three sides (SSS). Unlike the Law of Sines it has no ambiguous case: arccos returns a unique angle in [0, π], which is the whole range a triangle angle can occupy.
Area · 2
Area = ½ab sin C gives a triangle's area from two sides and the angle between them. It is the familiar ½ × base × height with b sin C standing in for the height. The angle must be the included one - the one between the two sides you used.
Heron's formula gives the area from the three sides alone: √(s(s−a)(s−b)(s−c)), where s = (a + b + c)/2 is the semiperimeter. No angle is needed, which makes it the one to use for SSS. If any factor comes out negative, the three lengths cannot form a triangle.
Inverse · 3
arcsin takes inputs in [−1, 1] and returns angles in [−π/2, π/2]. Sine repeats, so it has no true inverse until the range is restricted this way - which is why a calculator's arcsin never returns an obtuse angle, and why the Law of Sines' ambiguous case needs checking by hand.
arccos takes inputs in [−1, 1] and returns angles in [0, π]. That range covers every angle a triangle can have, which is why the Law of Cosines never suffers an ambiguous case while the Law of Sines does.
arctan accepts every real number and returns an angle in (−π/2, π/2). The interval is open, not closed: tangent has asymptotes at ±π/2, so those endpoints are approached but never reached.
Sum & Difference · 3
sin(α ± β) = sin α cos β ± cos α sin β. The signs match: a plus inside gives a plus outside. Use it to get exact values for angles you can build from known ones - sin 75° = sin(45° + 30°) - and to derive the double-angle formula by setting β = α.
cos(α ± β) = cos α cos β ∓ sin α sin β. The signs are opposite: a plus inside gives a minus outside. That sign flip is the most commonly dropped detail in the whole topic, and it comes from cosine being even while sine is odd.
tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β). The numerator takes the same sign as the angle sum and the denominator takes the opposite one. It is the formula behind the angle-between-two-lines result, where the tangents are the two slopes.
Double Angle · 3
sin 2θ = 2 sin θ cos θ, which drops straight out of the sine sum formula with β = α. It is the substitution that collapses a product of a sine and a cosine into a single term, and the one that makes ∫ sin θ cos θ dθ tractable.
cos 2θ has three equivalent forms: cos²θ − sin²θ, 2cos²θ − 1, and 1 − 2sin²θ. The Pythagorean identity converts between them. Pick whichever leaves you with only the function you already have - that choice is usually what makes a problem fall out in one line instead of five.
tan 2θ = 2 tan θ / (1 − tan²θ), from the tangent sum formula with β = α. It is undefined when tan²θ = 1, i.e. at odd multiples of π/4, which is exactly where 2θ lands on an asymptote of tangent.
Half Angle · 3
sin(θ/2) = ±√((1 − cos θ)/2), obtained by rearranging cos 2θ = 1 − 2sin²θ. The ± is not optional: you choose the sign from the quadrant θ/2 lands in, not from the sign of θ. Forgetting to check that quadrant is the classic way to lose the mark.
cos(θ/2) = ±√((1 + cos θ)/2), from rearranging cos 2θ = 2cos²θ − 1. Note the plus inside the radical, against the minus in the sine version. As with sine, the sign in front is decided by the quadrant of θ/2.
tan(θ/2) = (1 − cos θ)/sin θ = sin θ/(1 + cos θ). Unlike the sine and cosine half-angle formulas these need no ± and no square root - the sign takes care of itself. Use the first form when sin θ is comfortably non-zero and the second when cos θ is near −1.
Power Reduction · 2
sin²θ = (1 − cos 2θ)/2 trades a squared trig function for a plain cosine at double the angle. That is the whole point: a squared term cannot be integrated directly, but a cosine can, which makes this the standard first step for ∫ sin²θ dθ.
cos²θ = (1 + cos 2θ)/2 - the cosine counterpart of the sine power-reduction formula, differing only in the sign. Applied repeatedly it reduces any even power of sine or cosine down to first-degree terms.
Product to Sum · 3
sin α cos β = ½[sin(α + β) + sin(α − β)]. Products of trig functions are awkward to integrate and to graph; sums are not. This is also the identity behind acoustic beats, where two close frequencies multiply into a slow envelope.
cos α cos β = ½[cos(α − β) + cos(α + β)]. Both terms carry a plus sign here, unlike the sine-times-sine version. It comes straight from adding the cosine sum and difference formulas.
sin α sin β = ½[cos(α − β) − cos(α + β)]. Note that a product of two sines turns into cosines, and that the second term is subtracted - the one sign in the product-to-sum family worth double-checking.
Sum to Product · 4
sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2). Going from a sum to a product is what lets you solve equations by factoring: a product is zero when either factor is, so the whole solution set falls out at once.
sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2). The sine and cosine swap places compared with the sum version - the half-sum takes the cosine and the half-difference takes the sine.
cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2). Two cosines added give a product of two cosines, with no leading minus - which is what distinguishes it from the difference version.
cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2). The leading minus sign is the one people drop: a difference of cosines becomes a negative product of sines, not a positive one.