triangle, $\\sin\\theta=$ $\\text{[...]}$.","back":"For an angle $\\theta$ in a right-angled triangle, $\\sin\\theta=$ $\\boxed{\\frac{\\text{opposite}}{\\text{hypotenuse}}}$.","points":null,"mcqOptions":null},{"id":"56295209-5232-46b3-a78e-51a6f56029e1","cardType":"basic","front":"Which angle must be used with sides $a$ and $b$ in the triangle area formula?","back":"The included angle $C$ between sides $a$ and $b$.","points":1,"mcqOptions":null},{"id":"f2b25f3d-899f-4fc8-a24e-c5c60edae27f","cardType":"basic","front":"Under the usual triangle-labelling convention, where is side $a$ relative to angle $A$?","back":"Side $a$ is opposite angle $A$.","points":1,"mcqOptions":null},{"id":"be86c70a-1158-45ee-9c7d-28fb6e19e29f","cardType":"cloze","front":"If sides $a$ and $b$ enclose angle $C$, then $\\text{[...]}$.","back":"If sides $a$ and $b$ enclose angle $C$, then $\\boxed{\\text{Area}=\\frac12ab\\sin C}$.","points":null,"mcqOptions":null},{"id":"4d7c157a-10f1-442d-9ee0-e28370ed02fe","cardType":"basic","front":"What area formula uses sides $b$ and $c$?","back":"$\\text{Area}=\\frac12bc\\sin A$.","points":1,"mcqOptions":null},{"id":"0bf71b6d-90fb-4534-b6ec-8676627208fb","cardType":"basic","front":"What is wrong with pairing two sides with a non-included angle in the area formula?","back":"The substitution is invalid: the angle must lie between the sides.","points":1,"mcqOptions":null},{"id":"176c891d-9ff7-437b-9a74-81395cc11d64","cardType":"cloze","front":"Taking side $a$ as the base, the perpendicular height is $\\text{[...]}$.","back":"Taking side $a$ as the base, the perpendicular height is $\\boxed{h=b\\sin C}$.","points":null,"mcqOptions":null},{"id":"03291849-a855-4f97-b0c3-1d6ba1506978","cardType":"basic","front":"How does substituting $h=b\\sin C$ into $\\text{Area}=\\frac12ah$ produce the trigonometric area formula?","back":"It gives $\\text{Area}=\\frac12a(b\\sin C)=\\frac12ab\\sin C$.","points":2,"mcqOptions":null},{"id":"de5fa674-534c-43c0-b5b8-aaaef9ec62f0","cardType":"cloze","front":"The area formula remains valid for obtuse $C$ because $\\text{[...]}$.","back":"The area formula remains valid for obtuse $C$ because $\\boxed{\\sin(180^\\circ-C)=\\sin C}$.","points":null,"mcqOptions":null},{"id":"e7c99f56-a8f5-42b8-b7d7-2469d613d719","cardType":"basic","front":"Find the area of a triangle with sides 8 cm and 11 cm enclosing $42^\\circ$.","back":"$29.4\\text{ cm}^2$ to 3 significant figures.","points":2,"mcqOptions":null},{"id":"343de940-d98a-4be8-8308-3afde329a23c","cardType":"cloze","front":"The exact value needed for a $60^\\circ$ included angle is $\\text{[...]}$.","back":"The exact value needed for a $60^\\circ$ included angle is $\\boxed{\\sin60^\\circ=\\frac{\\sqrt3}{2}}$.","points":null,"mcqOptions":null},{"id":"109e8054-0348-4009-98b5-5917c21a7755","cardType":"basic","front":"Find the exact area enclosed by sides 7 cm and 12 cm with included angle $60^\\circ$.","back":"$21\\sqrt3\\text{ cm}^2$.","points":2,"mcqOptions":null},{"id":"eca6c3c9-64bf-457b-b06d-c043edef1be1","cardType":"basic","front":"How should an area involving an exact sine value be given when no decimal is requested?","back":"Leave the area in exact form.","points":1,"mcqOptions":null},{"id":"6d1039f3-c3d1-4840-be60-1d34186e77e9","cardType":"cloze","front":"Since $\\sin C\\leq1$, a triangle with enclosing sides $a$ and $b$ has maximum area $\\text{[...]}$.","back":"Since $\\sin C\\leq1$, a triangle with enclosing sides $a$ and $b$ has maximum area $\\boxed{\\frac12ab}$.","points":null,"mcqOptions":null},{"id":"46f99d7e-494d-45f0-a2c5-ff002eb3e820","cardType":"basic","front":"What does an area greater than $\\frac12ab$ indicate?","back":"A substitution or calculator-input error.","points":1,

1.4.3 Trigonometry Flashcards

1.4.3 Trigonometry Flashcards