8-5804-4e99-8eb3-bcd6ccc98386","topicSlug":"trigonometry-5","cards":[{"id":"f38c7227-c5b3-43dc-82f2-eeb1ec6decb2","cardType":"cloze","front":"On the unit circle, the point reached after turning through angle $x$ has coordinates $\\text{[...]}$.","back":"On the unit circle, the point reached after turning through angle $x$ has coordinates $\\boxed{(\\cos x,\\sin x)}$.","points":null,"mcqOptions":null},{"id":"a430ad26-8136-4e71-b274-aaf146f9cedc","cardType":"cloze","front":"An angle written using $\\pi$, such as $\\dfrac{\\pi}{3}$, is measured in [ ].","back":"An angle written using $\\pi$, such as $\\dfrac{\\pi}{3}$, is measured in radians.","points":null,"mcqOptions":null},{"id":"b47ee437-2e48-41c9-80eb-b06f53ed7ddb","cardType":"cloze","front":"Where $\\cos x\\neq 0$, the tangent identity is $\\text{[...]}$.","back":"Where $\\cos x\\neq 0$, the tangent identity is $\\boxed{\\tan x=\\dfrac{\\sin x}{\\cos x}}$.","points":null,"mcqOptions":null},{"id":"b7e8079d-634d-4424-a068-16408d9d34ec","cardType":"basic","front":"Why must $\\cos x\\neq 0$ when using $\\tan x=\\dfrac{\\sin x}{\\cos x}$?","back":"The fraction would otherwise involve division by zero.","points":2,"mcqOptions":null},{"id":"98cc89a9-1280-4495-a473-f3383295ccfc","cardType":"basic","front":"How do the right-triangle definitions produce the tangent identity?","back":"$\\dfrac{\\sin x}{\\cos x}=\\dfrac{\\text{opposite}}{\\text{adjacent}}=\\tan x$.","points":2,"mcqOptions":null},{"id":"79ff5f06-a1c3-4889-9d31-b6a44b0631c0","cardType":"cloze","front":"For every real $x$, the Pythagorean identity is $\\text{[...]}$.","back":"For every real $x$, the Pythagorean identity is $\\boxed{\\cos^2x+\\sin^2x=1}$.","points":null,"mcqOptions":null},{"id":"29f03f98-fc8c-4ea7-a9f4-0d71fee1d312","cardType":"basic","front":"How does Pythagoras on the unit circle give the Pythagorean identity?","back":"The legs are $\\cos x$ and $\\sin x$, so $\\cos^2x+\\sin^2x=1^2$.","points":2,"mcqOptions":null},{"id":"9c20e314-54d1-424e-9de4-d79e07549890","cardType":"basic","front":"Why does $\\cos^2x+\\sin^2x=1$ have no excluded real values?","back":"Sine and cosine are defined for every real angle.","points":2,"mcqOptions":null},{"id":"b9369968-eba0-4845-aba5-ef7191af5499","cardType":"cloze","front":"Rearranging the Pythagorean identity for $\\sin^2x$ gives $\\text{[...]}$.","back":"Rearranging the Pythagorean identity for $\\sin^2x$ gives $\\boxed{\\sin^2x=1-\\cos^2x}$.","points":null,"mcqOptions":null},{"id":"d70d3d4f-7d00-4563-8bd7-3632ad66c10f","cardType":"cloze","front":"Rearranging the Pythagorean identity for $\\cos^2x$ gives $\\text{[...]}$.","back":"Rearranging the Pythagorean identity for $\\cos^2x$ gives $\\boxed{\\cos^2x=1-\\sin^2x}$.","points":null,"mcqOptions":null},{"id":"5306db2d-d854-435f-b4e4-749a453bd74c","cardType":"basic","front":"Which substitution turns an equation containing $\\sin^2x$ and $\\cos x$ into a quadratic in $\\cos x$?","back":"Use $\\sin^2x=1-\\cos^2x$.","points":1,"mcqOptions":null},{"id":"e8f658a6-d05c-4031-bc4b-864adb0d47eb","cardType":"basic","front":"Which substitution turns an equation containing $\\cos^2x$ and $\\sin x$ into a quadratic in $\\sin x$?","back":"Use $\\cos^2x=1-\\sin^2x$.","points":1,"mcqOptions":null},{"id":"cdd019bd-cd35-4419-accb-3d33ae9df4fa","cardType":"basic","front":"Simplify $\\dfrac{1-\\cos^2x}{\\sin x}$ where the original expression is defined.","back":"$\\sin x$.","points":2,"mcqOptions":null},{"id":"ffb4bf80-2d64-4398-acbc-321490941e0b","cardType":"basic","front":"Which restriction must be retained after simplifying $\\dfrac{1-\\cos^2x}{\\sin x}$?","back":"$\\sin x\\neq 0$.","points":

1.5.4 Trigonometry Flashcards

1.5.4 Trigonometry Flashcards