{ "version": "2026.09.26-1", "checkedAt": "2026-09-26", "tolerance": { "rel": 1e-9, "abs": 1e-9 }, "samples": 800, "minValidSamples": 500, "counts": { "identities": 44, "examples": 28, "pythagorean": 3, "reciprocal-and-quotient": 5, "cofunction": 6, "sum-and-difference": 7, "product-to-sum": 4, "sum-to-product": 4, "triangle": 1, "double-angle": 5, "half-angle": 5, "power-reducing": 4 }, "verification": [ { "id": "pythagorean-1", "display": "sin²θ + cos²θ = 1", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "pythagorean-2", "display": "1 + tan²θ = sec²θ", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "pythagorean-3", "display": "1 + cot²θ = csc²θ", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "reciprocal-csc", "display": "csc θ = 1/sin θ", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "reciprocal-sec", "display": "sec θ = 1/cos θ", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "reciprocal-cot", "display": "cot θ = 1/tan θ", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "quotient-tan", "display": "tan θ = sin θ / cos θ", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "quotient-cot", "display": "cot θ = cos θ / sin θ", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "even-odd-sin", "display": "sin(−θ) = −sin θ", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "even-odd-cos", "display": "cos(−θ) = cos θ", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "even-odd-tan", "display": "tan(−θ) = −tan θ", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "cofunction-sin", "display": "sin θ = cos(90° − θ)", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "cofunction-cos", "display": "cos θ = sin(90° − θ)", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "cofunction-tan", "display": "tan θ = cot(90° − θ)", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "sum-sin", "display": "sin(A + B) = sin A cos B + cos A sin B", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "diff-sin", "display": "sin(A − B) = sin A cos B − cos A sin B", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "sum-cos", "display": "cos(A + B) = cos A cos B − sin A sin B", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "diff-cos", "display": "cos(A − B) = cos A cos B + sin A sin B", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "sum-tan", "display": "tan(A + B) = (tan A + tan B) / (1 − tan A tan B)", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "sum-tan-diff", "display": "tan(A − B) = (tan A − tan B) / (1 + tan A tan B)", "verdict": "equal", "tested": 800, "worstRel": 1.2303610563618922e-12 }, { "id": "sum-tan-alternate", "display": "tan A + tan B = sin(A + B) / (cos A cos B)", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "prod-sin-cos", "display": "sin A cos B = ½[sin(A + B) + sin(A − B)]", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "prod-cos-sin", "display": "cos A sin B = ½[sin(A + B) − sin(A − B)]", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "prod-cos-cos", "display": "cos A cos B = ½[cos(A − B) + cos(A + B)]", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "prod-sin-sin", "display": "sin A sin B = ½[cos(A − B) − cos(A + B)]", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "sum-prod-sin-sin", "display": "sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2)", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "sum-prod-sin-diff", "display": "sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2)", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "sum-prod-cos-cos", "display": "cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2)", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "sum-prod-cos-cos-diff", "display": "cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "triangle-area-sine", "display": "½ a b sin C = ½ b c sin A", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "double-sin", "display": "sin 2x = 2 sin x cos x", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "double-cos-1", "display": "cos 2x = cos²x − sin²x", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "double-cos-2", "display": "cos 2x = 2 cos²x − 1", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "double-cos-3", "display": "cos 2x = 1 − 2 sin²x", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "double-tan", "display": "tan 2x = 2 tan x / (1 − tan²x)", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "half-sin", "display": "sin(x/2) = ±√((1 − cos x)/2)", "verdict": "conditional-equal", "tested": 800, "worstRel": 0 }, { "id": "half-cos", "display": "cos(x/2) = ±√((1 + cos x)/2)", "verdict": "conditional-equal", "tested": 800, "worstRel": 0 }, { "id": "half-tan-a", "display": "tan(x/2) = (1 − cos x)/sin x", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "half-tan-b", "display": "tan(x/2) = sin x/(1 + cos x)", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "half-tan-c", "display": "tan(x/2) = csc x − cot x", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "power-sin-2", "display": "sin²x = (1 − cos 2x)/2", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "power-cos-2", "display": "cos²x = (1 + cos 2x)/2", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "power-sin-3", "display": "sin³x = (3 sin x − sin 3x)/4", "verdict": "equal", "tested": 800, "worstRel": 0 }, { "id": "power-cos-3", "display": "cos³x = (3 cos x + cos 3x)/4", "verdict": "equal", "tested": 800, "worstRel": 0 } ], "categories": [ { "id": "pythagorean", "name": "Pythagorean identities", "blurb": "sin²θ + cos²θ = 1 and the two versions you get by dividing it.", "summary": "The three Pythagorean identities come from one equation: divide sin²θ + cos²θ = 1 by sin²θ or by cos²θ and you get 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. Reach for them when a squared term has to disappear, when a 1 + (something)² shows up under a radical, or when you need to trade between sin/cos and tan/sec." }, { "id": "reciprocal-and-quotient", "name": "Reciprocal and quotient identities", "blurb": "csc, sec, cot as reciprocals; tan = sin/cos and cot = cos/sin.", "summary": "These six definitions are the translation layer: they turn tan, sec and csc into sin and cos, which is where every other identity lives. Most stalled verifications restart here — write everything in sines and cosines, then look again." }, { "id": "cofunction", "name": "Cofunction and even/odd identities", "blurb": "Complementary-angle pairs, and what happens to a negative angle.", "summary": "Cofunction identities say sin and cos are the same function read from the other acute angle of a right triangle: sin θ = cos(90° − θ). The even/odd identities say which functions flip sign when the angle does: cos and sec survive, sin, tan, cot and csc change sign." }, { "id": "sum-and-difference", "name": "Sum and difference identities", "blurb": "sin(a ± b), cos(a ± b), tan(a ± b) expanded.", "summary": "Six formulas that split one angle into two, or fuse two angles into one. They are the source that double-angle, half-angle and power-reducing formulas are all derived from, and they are what you use when an angle like 15° or 75° has to be computed without a calculator." }, { "id": "double-angle", "name": "Double-angle identities", "blurb": "sin 2x, the three forms of cos 2x, and tan 2x.", "summary": "Set b = a in the sum formulas. Cosine comes out in three equivalent shapes — cos²x − sin²x, 2cos²x − 1, 1 − 2sin²x — and picking the right one is half the work: the last two are what let you integrate cos²x or sin²x without a reduction formula." }, { "id": "half-angle", "name": "Half-angle identities", "blurb": "sin(x/2) and cos(x/2) — with the sign condition spelled out.", "summary": "Solve the double-angle formulas for the half angle. The square root is unavoidable, so the sign is a real question: cos(x/2) = +√((1+cos x)/2) holds for x in (−180°, 180°) and fails outside it. Our pages state the condition instead of hiding it." }, { "id": "product-to-sum", "name": "Product-to-sum identities", "blurb": "Products of sines and cosines rewritten as sums.", "summary": "Four formulas that turn sin A cos B and friends into sums. Integral calculus is the main customer: ∫sin 3x cos x dx is not approachable until the product is a sum, and the same trick handles ∫sin²x dx through cos²x = (1+cos 2x)/2." }, { "id": "sum-to-product", "name": "Sum-to-product identities", "blurb": "Sums and differences rewritten as a single product.", "summary": "The reverse direction. Solving sin A + sin B = 0 or factoring a trig expression is where these pay off, because an equation with one product equal to zero can be split into two simple equations." }, { "id": "power-reducing", "name": "Power-reducing identities", "blurb": "sin², cos², sin³, cos⁴ rewritten with no powers above one.", "summary": "Rearrangements of the double-angle formulas whose job is to lower a power. Integration almost always wants them: sin²x = (1 − cos 2x)/2 turns a squared trig term into something a first-year student can integrate in one line." }, { "id": "triangle", "name": "Triangle identities", "blurb": "The area formula ½ab sin C and what it implies via the sine rule.", "summary": "These are the identities where the variables are not free angles: A, B, C must be the angles of one triangle and a, b, c its opposite sides. Our sampler generates valid triangles instead of six random numbers, because random numbers make these formulas fail." } ], "triggers": [ { "id": "square-terms", "heading": "When your expression has squared terms", "question": "how to simplify sin squared plus cos squared", "explanation": "A squared sine or cosine is the single loudest signal in trigonometry. Either sin²θ + cos²θ = 1 kills the pair outright, or a power-reducing formula rewrites one of them without any power." }, { "id": "one-plus-square", "heading": "When you see 1 + (something)²", "question": "when to use 1 + tan squared = sec squared", "explanation": "1 + tan²θ, 1 + cot²θ and their csc/sec partners come straight from the Pythagorean family. Under a square root — √(1 + tan²θ) — this is usually the whole problem." }, { "id": "angle-sum", "heading": "When one angle is a sum or difference", "question": "when to use the sum formula for sine", "explanation": "sin(a + b), cos(a − b): expand when you need the pieces, collapse when you recognise the expansion. This is also how you compute sin 15° exactly." }, { "id": "double-angle", "heading": "When the angle is doubled and you want the single angle", "question": "when to use double angle identities", "explanation": "sin 2x, cos 2x, tan 2x in the question but x in the answers means: rewrite in terms of x. Cosine's three forms are the choice to make." }, { "id": "half-angle", "heading": "When the angle is halved, or a square root of 1 ± cos appears", "question": "when to use half angle formulas", "explanation": "√((1 − cos x)/2) is sin(x/2) with a sign condition. If the angle range is not given, the ± is part of the answer, not a footnote." }, { "id": "product", "heading": "When two different functions are multiplied", "question": "when to use product to sum", "explanation": "sin A cos B, cos A cos B, sin A sin B: a product of different functions rarely integrates, rarely solves, and always benefits from becoming a sum." }, { "id": "sum-of-same", "heading": "When the same function is added or subtracted", "question": "when to use sum to product", "explanation": "sin A + sin B, cos A − cos B: a sum of the same function becomes one product, which is what you need to factor an equation or evaluate a limit." }, { "id": "power", "heading": "When a power of sine or cosine has to be integrated", "question": "power reducing formula for sin squared", "explanation": "Any power above one is reduced with the power-reducing identities: sin², cos², sin³, cos³, sin⁴, cos⁴ each become a sum of first powers of multiple angles." }, { "id": "mixed-reciprocal", "heading": "When tan, sec, csc and cos are mixed together", "question": "how to verify a trig identity step by step", "explanation": "Mixed function families are the usual reason a verification stalls. Convert everything to sin and cos first, then apply the ordinary rules — this is the standard advice for a reason." }, { "id": "cofunction", "heading": "When the angles add up to 90°", "question": "cofunction identities", "explanation": "sin 32° and cos 58° are the same number. Complementary angles are what cofunction pairs describe." }, { "id": "negative-angle", "heading": "When a negative angle appears", "question": "sin of negative angle", "explanation": "Which functions change sign under x → −x decides a lot of homework answers. Cosine and secant are even; sine, tangent, cotangent and cosecant are odd." }, { "id": "triangle", "heading": "When the letters are the angles and sides of one triangle", "question": "area of a triangle with sine", "explanation": "a, b, c and A, B, C are not independent: A + B + C = 180° and the sides follow the sine rule. Any identity in this family is only true under those constraints." } ], "identitiesCountHint": 44, "files": [ "identities.json", "examples.json", "engine.json" ], "updatedNote": "Her kimlik rastgele örneklemeyle doğrulanır; sembolik ispat yerine geçmez." }