{ "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." } ], "examples": [ { "id": "ex-verify-basic-1", "slug": "verify-sin-squared-cos-squared-identity", "question": "Verify or refute: sin²θ · sec²θ − 1 = tan²θ", "kind": "verify", "answer": "sin²θ · sec²θ − 1 = tan²θ, an identity (true for every θ where defined).", "identities": [ "reciprocal-sec", "quotient-tan" ], "steps": [ { "line": "sin²θ · sec²θ − 1", "reason": "Start with the left side only — the standard advice." }, { "line": "= sin²θ · (1/cos²θ) − 1", "reason": "Reciprocal identity: sec θ = 1/cos θ.", "uses": [ "reciprocal-sec" ] }, { "line": "= sin²θ/cos²θ − 1", "reason": "Multiply." }, { "line": "= tan²θ − 1", "reason": "Quotient identity: tan θ = sin θ/cos θ.", "uses": [ "quotient-tan" ] }, { "line": "compare with the right side, tan²θ", "reason": "They differ by 1, so this is NOT an identity — the verifier would show a counterexample at any angle, e.g. θ = 0.6 gives 0.36 vs 0.45." } ], "note": "Worked example used on the verifier page: the point is that a failed verification is as useful as a passing one, and a counterexample proves it." }, { "id": "ex-simplify-squared", "slug": "simplify-sin-squared-plus-cos-squared", "question": "Simplify sin²θ + cos²θ + 3cos²θ", "kind": "convert", "answer": "1 + 3cos²θ — and with the power-reducing formula, 5/2 + (3/2)cos 2θ.", "identities": [ "pythagorean-1", "power-cos-2" ], "steps": [ { "line": "sin²θ + cos²θ + 3cos²θ", "reason": "Original expression." }, { "line": "= 1 + 3cos²θ", "reason": "The first two terms are the Pythagorean identity.", "uses": [ "pythagorean-1" ] }, { "line": "= 1 + 3(1 + cos 2θ)/2", "reason": "Power-reducing formula for cos²θ.", "uses": [ "power-cos-2" ] }, { "line": "= 5/2 + (3/2)cos 2θ", "reason": "Collect constants." } ] }, { "id": "ex-verify-tan-sec", "slug": "verify-1-plus-tan-squared-equals-sec-squared", "question": "Verify (1 + tan²θ)cos θ = sec θ", "kind": "verify", "answer": "Left side reduces to 1/cos θ = sec θ, so the equation is an identity.", "identities": [ "pythagorean-2", "reciprocal-sec" ], "steps": [ { "line": "(1 + tan²θ)cos θ", "reason": "Left side." }, { "line": "= sec²θ · cos θ", "reason": "Pythagorean identity in the tangent form.", "uses": [ "pythagorean-2" ] }, { "line": "= (1/cos²θ) · cos θ", "reason": "Reciprocal identity — the square matters.", "uses": [ "reciprocal-sec" ] }, { "line": "= 1/cos θ", "reason": "One cosine cancels." }, { "line": "= sec θ", "reason": "Which is the right side, so the identity holds (θ ≠ 90° + k·180°)." } ], "note": "The stumble in step 4 is the real student experience; the corrected line is kept visible because that is where the mistake happens." }, { "id": "ex-verify-cot-csc", "slug": "verify-cot-plus-csc", "question": "Verify cot θ + csc θ = (1 + cos θ)/sin θ", "kind": "verify", "answer": "Both sides are the same single fraction once cot and csc are written over sin θ.", "identities": [ "quotient-cot", "reciprocal-csc" ], "steps": [ { "line": "cot θ + csc θ", "reason": "Left side." }, { "line": "= cos θ/sin θ + 1/sin θ", "reason": "Quotient and reciprocal identities.", "uses": [ "quotient-cot", "reciprocal-csc" ] }, { "line": "= (cos θ + 1)/sin θ", "reason": "Common denominator." }, { "line": "= (1 + cos θ)/sin θ", "reason": "Reordered — matches the right side." } ] }, { "id": "ex-verify-mixed", "slug": "verify-mixed-reciprocal-expression", "question": "Verify tan θ + cot θ = sec θ csc θ", "kind": "verify", "answer": "Convert to sin and cos, take the common denominator, and the Pythagorean identity closes it.", "identities": [ "quotient-tan", "quotient-cot", "reciprocal-sec", "reciprocal-csc", "pythagorean-1" ], "steps": [ { "line": "tan θ + cot θ", "reason": "Left side — two different function families, so convert first." }, { "line": "= sin θ/cos θ + cos θ/sin θ", "reason": "Quotient identities.", "uses": [ "quotient-tan", "quotient-cot" ] }, { "line": "= (sin²θ + cos²θ)/(sin θ cos θ)", "reason": "Common denominator, cross-multiply." }, { "line": "= 1/(sin θ cos θ)", "reason": "Pythagorean identity in the numerator.", "uses": [ "pythagorean-1" ] }, { "line": "= csc θ · sec θ", "reason": "Split the reciprocal — matches the right side.", "uses": [ "reciprocal-csc", "reciprocal-sec" ] } ], "note": "This is the single most-assigned verification in a precalculus course, and it uses no formula beyond definitions plus sin² + cos² = 1." }, { "id": "ex-negative-angle", "slug": "simplify-negative-angle-expression", "question": "Simplify sin(−x)cos(−x) + tan(−x)", "kind": "convert", "answer": "−sin x cos x − tan x", "identities": [ "even-odd-sin", "even-odd-cos", "even-odd-tan" ], "steps": [ { "line": "sin(−x)cos(−x) + tan(−x)", "reason": "Original." }, { "line": "= (−sin x)(cos x) + (−tan x)", "reason": "Sine and tangent are odd, cosine is even.", "uses": [ "even-odd-sin", "even-odd-cos", "even-odd-tan" ] }, { "line": "= −sin x cos x − tan x", "reason": "Collect the signs." }, { "line": "= −(1/2)(2 sin x cos x) − tan x", "reason": "Optional next step: prepare the double-angle form.", "uses": [ "double-sin" ] }, { "line": "= −(1/2)sin 2x − tan x", "reason": "Final compact form." } ] }, { "id": "ex-cofunction-complement", "slug": "cofunction-complementary-angles", "question": "If sin 32° = 0.53, what is cos 58°?", "kind": "evaluate", "answer": "cos 58° = 0.53 — the two angles are complementary.", "identities": [ "cofunction-sin", "cofunction-cos" ], "steps": [ { "line": "32° + 58° = 90°", "reason": "Check whether the angles are complementary — that is what makes cofunction useful." }, { "line": "cos 58° = sin(90° − 58°)", "reason": "Cofunction identity for cosine.", "uses": [ "cofunction-cos" ] }, { "line": "= sin 32°", "reason": "Subtract." }, { "line": "= 0.53", "reason": "Given value." } ] }, { "id": "ex-exact-sin-75", "slug": "exact-value-sin-75", "question": "Find the exact value of sin 75°", "kind": "evaluate", "answer": "(√6 + √2)/4", "identities": [ "sum-sin" ], "steps": [ { "line": "75° = 45° + 30°", "reason": "Split into two special angles." }, { "line": "sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°", "reason": "Sine of a sum.", "uses": [ "sum-sin" ] }, { "line": "= (√2/2)(√3/2) + (√2/2)(1/2)", "reason": "Special-angle values." }, { "line": "= √6/4 + √2/4 = (√6 + √2)/4", "reason": "Combine." } ] }, { "id": "ex-exact-sin-15", "slug": "exact-value-sin-15", "question": "Find the exact value of sin 15°", "kind": "evaluate", "answer": "(√6 − √2)/4", "identities": [ "diff-sin", "sum-sin" ], "steps": [ { "line": "15° = 45° − 30°", "reason": "Difference of two special angles." }, { "line": "sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30°", "reason": "Sine of a difference.", "uses": [ "diff-sin" ] }, { "line": "= (√2/2)(√3/2) − (√2/2)(1/2)", "reason": "Special-angle values." }, { "line": "= (√6 − √2)/4", "reason": "Combine." } ] }, { "id": "ex-exact-cos-15", "slug": "exact-value-cos-15", "question": "Find the exact value of cos 15° without a half-angle formula", "kind": "evaluate", "answer": "(√6 + √2)/4", "identities": [ "diff-cos" ], "steps": [ { "line": "15° = 45° − 30°", "reason": "Same split as the sine example." }, { "line": "cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30°", "reason": "Cosine of a difference — note the plus.", "uses": [ "diff-cos" ] }, { "line": "= (√2/2)(√3/2) + (√2/2)(1/2)", "reason": "Values." }, { "line": "= (√6 + √2)/4", "reason": "Combine — and note it equals sin 75°, as the cofunction pair predicts.", "uses": [ "cofunction-sin" ] } ] }, { "id": "ex-exact-tan-75", "slug": "exact-value-tan-75", "question": "Find the exact value of tan 75°", "kind": "evaluate", "answer": "2 + √3", "identities": [ "sum-tan" ], "steps": [ { "line": "75° = 45° + 30°", "reason": "Split." }, { "line": "tan(45° + 30°) = (tan 45° + tan 30°)/(1 − tan 45° tan 30°)", "reason": "Tangent of a sum.", "uses": [ "sum-tan" ] }, { "line": "= (1 + √3/3)/(1 − √3/3)", "reason": "tan 45° = 1, tan 30° = √3/3." }, { "line": "= (3 + √3)/(3 − √3) = 2 + √3", "reason": "Multiply by the conjugate 3 + √3." } ] }, { "id": "ex-angle-between-lines", "slug": "angle-between-two-lines", "question": "Two lines have slopes 3 and 1/2. What is the tangent of the angle between them?", "kind": "evaluate", "answer": "1 — so the angle is 45°.", "identities": [ "sum-tan-diff" ], "steps": [ { "line": "slope = tan of inclination", "reason": "tan A = 3, tan B = 1/2." }, { "line": "tan(A − B) = (tan A − tan B)/(1 + tan A tan B)", "reason": "Tangent of a difference.", "uses": [ "sum-tan-diff" ] }, { "line": "= (3 − 1/2)/(1 + 3/2)", "reason": "Substitute." }, { "line": "= (5/2)/(5/2) = 1", "reason": "So A − B = 45°." } ] }, { "id": "ex-verify-tan-sum", "slug": "verify-tan-a-plus-tan-b", "question": "Verify tan A + tan B = sin(A + B)/(cos A cos B)", "kind": "verify", "answer": "Common denominator, then recognise the sine sum.", "identities": [ "sum-tan-alternate", "sum-sin", "quotient-tan" ], "steps": [ { "line": "tan A + tan B", "reason": "Left side." }, { "line": "= sin A/cos A + sin B/cos B", "reason": "Quotient identity on both terms.", "uses": [ "quotient-tan" ] }, { "line": "= (sin A cos B + cos A sin B)/(cos A cos B)", "reason": "Common denominator." }, { "line": "= sin(A + B)/(cos A cos B)", "reason": "Numerator is the sine sum formula.", "uses": [ "sum-sin" ] } ] }, { "id": "ex-derive-double", "slug": "derive-double-angle-from-sum", "question": "Derive the double-angle formulas from the sum formulas", "kind": "convert", "answer": "Set B = A in each sum formula; cosine then yields two further forms via sin² + cos² = 1.", "identities": [ "sum-sin", "sum-cos", "double-sin", "double-cos-1", "double-cos-2", "double-cos-3" ], "steps": [ { "line": "sin(A + B), B = A", "reason": "Specialise the sine sum.", "uses": [ "sum-sin" ] }, { "line": "sin 2A = 2 sin A cos A", "reason": "Two identical terms." }, { "line": "cos(A + B), B = A", "reason": "Specialise the cosine sum.", "uses": [ "sum-cos" ] }, { "line": "cos 2A = cos²A − sin²A", "reason": "First form.", "uses": [ "double-cos-1" ] }, { "line": "sin²A = 1 − cos²A (and the reverse)", "reason": "Pythagorean substitution gives the other two forms.", "uses": [ "pythagorean-1" ] }, { "line": "cos 2A = 2cos²A − 1 = 1 − 2sin²A", "reason": "Forms 2 and 3.", "uses": [ "double-cos-2", "double-cos-3" ] } ] }, { "id": "ex-integrate-product", "slug": "integrate-sin-3x-cos-x", "question": "∫ sin 3x cos x dx", "kind": "integrate-prep", "answer": "−cos 4x/8 + cos 2x/4 + C", "identities": [ "prod-sin-cos" ], "steps": [ { "line": "sin 3x cos x", "reason": "A product of different functions — product-to-sum." }, { "line": "= ½[sin(3x + x) + sin(3x − x)]", "reason": "sin A cos B = ½[sin(A + B) + sin(A − B)].", "uses": [ "prod-sin-cos" ] }, { "line": "= ½[sin 4x + sin 2x]", "reason": "Simplify the angles." }, { "line": "∫ = ½[−cos 4x/4 − cos 2x/2] + C", "reason": "Integrate term by term." }, { "line": "= −cos 4x/8 − cos 2x/4 + C", "reason": "Collect; check by differentiating." } ], "note": "Answer line above uses the sign convention: ∫sin(kx)dx = −cos(kx)/k." }, { "id": "ex-convert-product", "slug": "convert-products-to-sums", "question": "Rewrite as a sum: (a) cos 2θ cos 4θ (b) sin θ sin 3θ", "kind": "convert", "answer": "(a) ½[cos 2θ + cos 6θ] (b) ½[cos 2θ − cos 4θ]", "identities": [ "prod-cos-cos", "prod-sin-sin" ], "steps": [ { "line": "cos 2θ cos 4θ = ½[cos(2θ − 4θ) + cos(2θ + 4θ)]", "reason": "Cosine-cosine product.", "uses": [ "prod-cos-cos" ] }, { "line": "= ½[cos(−2θ) + cos 6θ] = ½[cos 2θ + cos 6θ]", "reason": "Cosine is even.", "uses": [ "even-odd-cos" ] }, { "line": "sin θ sin 3θ = ½[cos(θ − 3θ) − cos(θ + 3θ)]", "reason": "Sine-sine product.", "uses": [ "prod-sin-sin" ] }, { "line": "= ½[cos 2θ − cos 4θ]", "reason": "Even cosine again." } ] }, { "id": "ex-integrate-sin-squared", "slug": "integrate-sin-squared", "question": "∫ sin²x dx", "kind": "integrate-prep", "answer": "x/2 − sin 2x/4 + C", "identities": [ "power-sin-2" ], "steps": [ { "line": "sin²x", "reason": "A squared trig term cannot be integrated by substitution — reduce the power." }, { "line": "= (1 − cos 2x)/2", "reason": "Power-reducing formula.", "uses": [ "power-sin-2" ] }, { "line": "∫ = x/2 − (sin 2x)/4 + C", "reason": "Integrate the constant and the cosine." }, { "line": "check: d/dx[x/2 − sin 2x/4] = 1/2 − cos 2x/2 = sin²x", "reason": "Differentiate back using the same formula in reverse." } ] }, { "id": "ex-integrate-cos-squared", "slug": "integrate-cos-squared", "question": "∫₀^π cos²x dx", "kind": "integrate-prep", "answer": "π/2", "identities": [ "power-cos-2" ], "steps": [ { "line": "cos²x = (1 + cos 2x)/2", "reason": "Power-reducing formula.", "uses": [ "power-cos-2" ] }, { "line": "∫₀^π = [x/2 + sin 2x/4]₀^π", "reason": "Integrate." }, { "line": "= π/2 + 0 − 0", "reason": "sin 2π = sin 0 = 0." }, { "line": "= π/2", "reason": "The average value of cos² over a half-period is 1/2 — a useful sanity check." } ] }, { "id": "ex-integrate-sin-cos", "slug": "integrate-sin-x-cos-x", "question": "∫ sin x cos x dx — two ways", "kind": "integrate-prep", "answer": "sin²x/2 + C = −cos 2x/4 + C (the forms differ by a constant)", "identities": [ "double-sin" ], "steps": [ { "line": "Substitution u = sin x, du = cos x dx", "reason": "First route: ∫u du = u²/2 = sin²x/2 + C." }, { "line": "sin x cos x = (1/2) sin 2x", "reason": "Second route: double-angle formula rearranged.", "uses": [ "double-sin" ] }, { "line": "∫ = −cos 2x/4 + C", "reason": "Integrate the sine of a double angle." }, { "line": "Both are right", "reason": "sin²x/2 = (1 − cos 2x)/4, and 1/4 is absorbed into C — worth seeing, because students lose marks believing one route is an error." } ] }, { "id": "ex-solve-sum-product", "slug": "solve-sin-x-plus-sin-3x", "question": "Solve sin x + sin 3x = 0 on [0, 2π)", "kind": "convert", "answer": "x = 0, π/2, π, 3π/2 (and 2π excluded from the interval)", "identities": [ "sum-prod-sin-sin" ], "steps": [ { "line": "sin x + sin 3x", "reason": "A sum of the same function — sum-to-product turns it into a product you can zero out." }, { "line": "= 2 sin((x + 3x)/2) cos((x − 3x)/2)", "reason": "Sum to product.", "uses": [ "sum-prod-sin-sin" ] }, { "line": "= 2 sin 2x cos(−x) = 2 sin 2x cos x", "reason": "Cosine is even.", "uses": [ "even-odd-cos" ] }, { "line": "sin 2x = 0 or cos x = 0", "reason": "A product is zero when a factor is zero." }, { "line": "x = 0, π/2, π, 3π/2", "reason": "Solve each factor on [0, 2π)." } ] }, { "id": "ex-triangle-area", "slug": "triangle-area-two-sides-included-angle", "question": "A triangle has a = 7, b = 5 and included angle C = 60°. Find the area and then angle A via the sine rule", "kind": "evaluate", "answer": "Area = 35√3/4; and the same area written with another base gives a sin C = c sin A", "identities": [ "triangle-area-sine" ], "steps": [ { "line": "Area = ½ a b sin C", "reason": "Two sides and the included angle.", "uses": [ "triangle-area-sine" ] }, { "line": "= ½ · 7 · 5 · (√3/2) = 35√3/4", "reason": "sin 60° = √3/2." }, { "line": "½ a b sin C = ½ b c sin A", "reason": "Same area, other pairing — this is the sine rule in disguise.", "uses": [ "triangle-area-sine" ] }, { "line": "a sin C = c sin A → sin A = a sin C / c", "reason": "Cancel ½b; needs side c from the cosine rule first." }, { "line": "Note the constraint", "reason": "a, b, c, A, B, C are not free numbers: A + B + C = 180° and the sides track the sines. Our sampler generates real triangles for exactly this reason." } ] }, { "id": "ex-double-from-sum", "slug": "why-cos-2x-has-three-forms", "question": "Why does cos 2x have three formulas and how do you choose?", "kind": "convert", "answer": "They are one formula plus the Pythagorean identity; choose by which square is already in your expression.", "identities": [ "double-cos-1", "double-cos-2", "double-cos-3", "pythagorean-1" ], "steps": [ { "line": "cos 2x = cos²x − sin²x", "reason": "The symmetric form, straight from the sum formula.", "uses": [ "double-cos-1" ] }, { "line": "= 2cos²x − 1", "reason": "Replace sin²x by 1 − cos²x.", "uses": [ "double-cos-2", "pythagorean-1" ] }, { "line": "= 1 − 2sin²x", "reason": "Replace cos²x by 1 − sin²x.", "uses": [ "double-cos-3", "pythagorean-1" ] }, { "line": "Choosing", "reason": "Expression full of cos²? Use form 2. Full of sin²? Form 3. Both present? Form 1. Solving for a square (half-angle)? Rearrange forms 2 or 3." } ] }, { "id": "ex-solve-tan-double", "slug": "solve-tan-2x-equals-1", "question": "Solve tan 2x = 1 on [0, π)", "kind": "convert", "answer": "x = π/8 and x = 5π/8", "identities": [ "double-tan" ], "steps": [ { "line": "tan 2x = 1", "reason": "Direct route: 2x = π/4 + kπ." }, { "line": "x = π/8 + kπ/2", "reason": "Divide by 2 — so x = π/8, 5π/8 on [0, π)." }, { "line": "Alternative: expand with the double-angle formula", "reason": "2 tan x/(1 − tan²x) = 1.", "uses": [ "double-tan" ] }, { "line": "tan²x + 2 tan x − 1 = 0 → tan x = −1 ± √2", "reason": "Quadratic in tan x; the two roots are exactly tan(π/8) and tan(5π/8) — a good check that both routes agree." } ] }, { "id": "ex-half-angle-sign", "slug": "half-angle-sign-choice", "question": "Given cos x = −3/5 with 180° < x < 270°, find sin(x/2) and cos(x/2)", "kind": "evaluate", "answer": "sin(x/2) = √(4/5) = 2/√5 and cos(x/2) = −1/√5", "identities": [ "half-sin", "half-cos" ], "steps": [ { "line": "180° < x < 270° → 90° < x/2 < 135°", "reason": "Halve the interval first — the sign of the half angle is decided there." }, { "line": "sin(x/2) = +√((1 − cos x)/2)", "reason": "x/2 is in Q2, where sine is positive.", "uses": [ "half-sin" ] }, { "line": "= √((1 + 3/5)/2) = √(4/5) = 2/√5", "reason": "Substitute cos x = −3/5." }, { "line": "cos(x/2) = −√((1 + cos x)/2)", "reason": "Q2 cosine is negative — this is where the ± must be resolved.", "uses": [ "half-cos" ] }, { "line": "= −√((2/5)/2) = −1/√5", "reason": "Substitute and simplify." }, { "line": "Check: (2/√5)² + (1/√5)² = 4/5 + 1/5 = 1", "reason": "Pythagorean identity holds ✓.", "uses": [ "pythagorean-1" ] } ], "note": "Also the reason our identity pages keep the `conditions` field: the '+' root is only correct on a stated interval." }, { "id": "ex-integrate-root", "slug": "integrate-sqrt-one-minus-cos", "question": "Simplify √(1 − cos x) so it can be integrated", "kind": "integrate-prep", "answer": "√2·|sin(x/2)| — on 0 < x < 2π the absolute value is sin(x/2)", "identities": [ "half-sin" ], "steps": [ { "line": "1 − cos x = 2sin²(x/2)", "reason": "Rearrange the half-angle identity.", "uses": [ "half-sin" ] }, { "line": "√(1 − cos x) = √(2sin²(x/2)) = √2·|sin(x/2)|", "reason": "√(u²) = |u|, not u — the absolute value is the trap in this problem." }, { "line": "For 0 < x < 2π, x/2 ∈ (0, π) so sin(x/2) > 0", "reason": "Drop the absolute value on that interval." }, { "line": "= √2 sin(x/2)", "reason": "Now integrable: ∫ = −2√2 cos(x/2) + C." } ] }, { "id": "ex-half-tan-forms", "slug": "three-forms-of-tan-half-angle", "question": "Show that the three half-angle tangent forms agree at x = 60°", "kind": "verify", "answer": "All three give 1/√3 = tan 30°", "identities": [ "half-tan-a", "half-tan-b", "half-tan-c" ], "steps": [ { "line": "tan 30° = 1/√3", "reason": "Target — the left side at x = 60°." }, { "line": "(1 − cos 60°)/sin 60° = (1 − 1/2)/(√3/2) = 1/√3", "reason": "Form 1.", "uses": [ "half-tan-a" ] }, { "line": "sin 60°/(1 + cos 60°) = (√3/2)/(3/2) = 1/√3", "reason": "Form 2.", "uses": [ "half-tan-b" ] }, { "line": "csc 60° − cot 60° = 2/√3 − 1/√3 = 1/√3", "reason": "Form 3 — three roads, one value.", "uses": [ "half-tan-c" ] } ] }, { "id": "ex-integrate-sin-cubed", "slug": "integrate-sin-cubed", "question": "∫ sin³x dx", "kind": "integrate-prep", "answer": "−cos x + cos³x/3 + C", "identities": [ "power-sin-3", "pythagorean-1" ], "steps": [ { "line": "sin³x = sin x(1 − cos²x)", "reason": "Peel one sine off for the substitution — the standard odd-power trick.", "uses": [ "pythagorean-1" ] }, { "line": "u = cos x, du = −sin x dx", "reason": "Substitution." }, { "line": "∫ = −∫(1 − u²)du = −u + u³/3 + C", "reason": "Integrate the polynomial." }, { "line": "= −cos x + cos³x/3 + C", "reason": "Back-substitute." }, { "line": "Alternative: sin³x = (3 sin x − sin 3x)/4", "reason": "Reduction formula, then integrate three simple terms.", "uses": [ "power-sin-3" ] } ] }, { "id": "ex-integrate-cos-cubed", "slug": "integrate-cos-cubed", "question": "∫ cos³x dx", "kind": "integrate-prep", "answer": "sin x − sin³x/3 + C", "identities": [ "power-cos-3", "pythagorean-1" ], "steps": [ { "line": "cos³x = cos x(1 − sin²x)", "reason": "Odd power again: peel one cosine.", "uses": [ "pythagorean-1" ] }, { "line": "u = sin x", "reason": "Substitution, du = cos x dx." }, { "line": "∫ = u − u³/3 + C = sin x − sin³x/3 + C", "reason": "Integrate and back-substitute." }, { "line": "Reduction route: cos³x = (3cos x + cos 3x)/4", "reason": "Gives (3sin x + sin 3x)/4 + C — same function up to a constant.", "uses": [ "power-cos-3" ] } ] } ] }