{
  "unit": 11,
  "title": "Trigonometric Identities, Equations & Oblique Triangles",
  "terms": [
    {
      "en": "Pythagorean identity",
      "es": "identidad pitagórica",
      "fr": "identité pythagoricienne",
      "ar": "متطابقة فيثاغورس المثلثية",
      "pt": "identidade pitagórica",
      "definition": "The identity sin²θ+cos²θ=1 for all real θ."
    },
    {
      "en": "reciprocal identity",
      "es": "identidad recíproca",
      "fr": "identité réciproque",
      "ar": "متطابقة المقلوب",
      "pt": "identidade recíproca",
      "definition": "An identity such as secθ=1/cosθ wherever defined."
    },
    {
      "en": "angle sum formula",
      "es": "fórmula de suma de ángulos",
      "fr": "formule de somme des angles",
      "ar": "صيغة مجموع زاويتين",
      "pt": "fórmula da soma de ângulos",
      "definition": "A formula expressing sin(a+b) or cos(a+b) in terms of a and b."
    },
    {
      "en": "identity verification",
      "es": "verificación de identidades",
      "fr": "vérification des identités",
      "ar": "التحقق من المتطابقات",
      "pt": "verificação de identidades",
      "definition": "Transforming one side into the other with valid algebraic identities."
    },
    {
      "en": "trigonometric identity",
      "es": "identidad trigonométrica",
      "fr": "identité trigonométrique",
      "ar": "متطابقة مثلثية",
      "pt": "identidade trigonométrica",
      "definition": "An equality true for every input common to both sides' domains."
    },
    {
      "en": "sum identity",
      "es": "identidad de suma",
      "fr": "identité de somme",
      "ar": "متطابقة المجموع",
      "pt": "identidade de soma",
      "definition": "A trigonometric equality relating a sum of angles to separate terms."
    },
    {
      "en": "difference identity",
      "es": "identidad de diferencia",
      "fr": "identité de différence",
      "ar": "متطابقة الفرق",
      "pt": "identidade de diferença",
      "definition": "A trigonometric equality for expressions involving a-b."
    },
    {
      "en": "double-angle identity",
      "es": "identidad del ángulo doble",
      "fr": "identité d'angle double",
      "ar": "متطابقة الزاوية المضاعفة",
      "pt": "identidade do ângulo duplo",
      "definition": "An identity relating functions of 2θ to functions of θ."
    },
    {
      "en": "half-angle identity",
      "es": "identidad del ángulo mitad",
      "fr": "identité de demi-angle",
      "ar": "متطابقة نصف الزاوية",
      "pt": "identidade do meio ângulo",
      "definition": "An identity relating functions of θ/2 to functions of θ with sign restrictions."
    },
    {
      "en": "trigonometric equation",
      "es": "ecuación trigonométrica",
      "fr": "équation trigonométrique",
      "ar": "معادلة مثلثية",
      "pt": "equação trigonométrica",
      "definition": "An equation whose solutions are angles satisfying a trigonometric relation."
    },
    {
      "en": "principal value",
      "es": "valor principal",
      "fr": "valeur principale",
      "ar": "القيمة الرئيسية",
      "pt": "valor principal",
      "definition": "The output chosen from a restricted interval by an inverse trig function."
    },
    {
      "en": "general solution",
      "es": "solución general",
      "fr": "solution générale",
      "ar": "الحل العام",
      "pt": "solução geral",
      "definition": "All solutions described using periodicity and integer parameters."
    },
    {
      "en": "law of sines",
      "es": "ley de los senos",
      "fr": "loi des sinus",
      "ar": "قانون الجيوب",
      "pt": "lei dos senos",
      "definition": "For a triangle a/sin A=b/sin B=c/sin C."
    },
    {
      "en": "ambiguous SSA case",
      "es": "caso ambiguo LLA",
      "fr": "cas ambigu SSA",
      "ar": "الحالة الملتبسة ضلع ضلع زاوية",
      "pt": "caso ambíguo LLA",
      "definition": "Two sides and a nonincluded angle may determine zero, one, or two triangles."
    },
    {
      "en": "triangle altitude",
      "es": "altura del triángulo",
      "fr": "hauteur du triangle",
      "ar": "ارتفاع المثلث",
      "pt": "altura do triângulo",
      "definition": "A perpendicular segment from a vertex to the opposite side's line."
    },
    {
      "en": "law of cosines",
      "es": "ley de los cosenos",
      "fr": "loi des cosinus",
      "ar": "قانون جيب التمام",
      "pt": "lei dos cossenos",
      "definition": "For opposite side c, c²=a²+b²-2ab cos C."
    },
    {
      "en": "Heron formula",
      "es": "fórmula de Herón",
      "fr": "formule de Héron",
      "ar": "صيغة هيرون",
      "pt": "fórmula de Heron",
      "definition": "For semiperimeter s, area=sqrt(s(s-a)(s-b)(s-c))."
    },
    {
      "en": "triangle area",
      "es": "área del triángulo",
      "fr": "aire du triangle",
      "ar": "مساحة المثلث",
      "pt": "área do triângulo",
      "definition": "Half the product of two sides and the sine of their included angle."
    },
    {
      "en": "navigation bearing",
      "es": "rumbo de navegación",
      "fr": "relèvement de navigation",
      "ar": "اتجاه ملاحي",
      "pt": "rumo de navegação",
      "definition": "A direction angle based on a specified compass convention."
    },
    {
      "en": "vector resolution",
      "es": "descomposición vectorial",
      "fr": "décomposition vectorielle",
      "ar": "تحليل المتجه إلى مركبات",
      "pt": "decomposição vetorial",
      "definition": "Separating a vector into coordinate-direction components."
    },
    {
      "en": "multi-step trig model",
      "es": "modelo trigonométrico de varios pasos",
      "fr": "modèle trigonométrique à étapes",
      "ar": "نموذج مثلثي متعدد الخطوات",
      "pt": "modelo trigonométrico em etapas",
      "definition": "A problem requiring several justified trigonometric relationships and checks."
    }
  ],
  "schema_version": "1.1"
}