Flashcards","topicId":"c3c922f9-eacf-4120-b39a-12f910e243f3","topicSlug":"trigonometry-14","cards":[{"id":"f10b03d8-a78a-4921-9f6b-0c59c9076f9c","cardType":"basic","front":"What amplitude must be used when combining $p\\cos x+q\\sin x$?","back":"$R=\\sqrt{p^2+q^2}$, with $R>0$.","points":1,"mcqOptions":null},{"id":"9172910f-c00a-45a6-b123-8e7c8aba0d2a","cardType":"basic","front":"What does the phase angle $\\alpha$ control in $R\\cos(x-\\alpha)$?","back":"The graph's horizontal shift.","points":1,"mcqOptions":null},{"id":"a1937e66-73c6-447f-81da-e6fd858a0b6c","cardType":"cloze","front":"The compound-angle identity is $\\text{[...]}$.","back":"The compound-angle identity is $\\boxed{\\cos(x-\\alpha)=\\cos x\\cos\\alpha+\\sin x\\sin\\alpha}$.","points":null,"mcqOptions":null},{"id":"ce91c64c-7a55-4173-86ee-ccf508502fb3","cardType":"cloze","front":"The compound-angle identity is $\\text{[...]}$.","back":"The compound-angle identity is $\\boxed{\\sin(x+\\alpha)=\\sin x\\cos\\alpha+\\cos x\\sin\\alpha}$.","points":null,"mcqOptions":null},{"id":"9d558997-3d25-478f-975e-1bfeebcacf9a","cardType":"cloze","front":"The compound-angle identity producing a negative sine term is $\\text{[...]}$.","back":"The compound-angle identity producing a negative sine term is $\\boxed{\\cos(x+\\alpha)=\\cos x\\cos\\alpha-\\sin x\\sin\\alpha}$.","points":null,"mcqOptions":null},{"id":"abb64b24-9497-4df0-ade9-d07c7849bdda","cardType":"basic","front":"What is the maximum value of $p\\cos x+q\\sin x$?","back":"$\\sqrt{p^2+q^2}$.","points":1,"mcqOptions":null},{"id":"afe269e2-cf74-4ed6-81b3-3cf10a07a810","cardType":"basic","front":"What is the minimum value of $p\\cos x+q\\sin x$?","back":"$-\\sqrt{p^2+q^2}$.","points":1,"mcqOptions":null},{"id":"9a8bde95-88b8-4292-9647-66beed75ab7d","cardType":"cloze","front":"The range of $p\\cos x+q\\sin x$ is $\\text{[...]}$.","back":"The range of $p\\cos x+q\\sin x$ is $\\boxed{-R\\leq p\\cos x+q\\sin x\\leq R}$.","points":null,"mcqOptions":null},{"id":"3115adb5-44ce-4978-ab5f-d2948d12bb35","cardType":"basic","front":"What is the maximum of $c+p\\cos x+q\\sin x$?","back":"$c+R$, where $R=\\sqrt{p^2+q^2}$.","points":1,"mcqOptions":null},{"id":"e53f3267-5c37-4e97-bc15-91493fe1a8bd","cardType":"basic","front":"What is the minimum of $c+p\\cos x+q\\sin x$?","back":"$c-R$, where $R=\\sqrt{p^2+q^2}$.","points":1,"mcqOptions":null},{"id":"f07e9d3c-6bce-4fa1-a90f-396043ac74ec","cardType":"basic","front":"Which acute-angle cosine form suits $p\\cos x+q\\sin x$ when $p>0$ and $q>0$?","back":"$R\\cos(x-\\alpha)$.","points":1,"mcqOptions":null},{"id":"06bb2a88-c720-4fe9-8803-820b24e8d90f","cardType":"basic","front":"Which acute-angle cosine form suits $p\\cos x+q\\sin x$ when $p>0$ and $q<0$?","back":"$R\\cos(x+\\alpha)$.","points":1,"mcqOptions":null},{"id":"443846f5-12e4-41b4-9565-afed5d88d4e4","cardType":"cloze","front":"For $p\\cos x+q\\sin x=R\\cos(x-\\alpha)$, coefficient matching gives $\\text{[...]}$.","back":"For $p\\cos x+q\\sin x=R\\cos(x-\\alpha)$, coefficient matching gives $\\boxed{\\tan\\alpha=\\frac{q}{p}}$.","points":null,"mcqOptions":null},{"id":"f39e5be1-42fa-444a-95f5-cc91bfa379fc","cardType":"cloze","front":"For $p\\cos x+q\\sin x=R\\sin(x+\\alpha)$, coefficient matching gives $\\text{[...]}$.","back":"For $p\\cos x+q\\sin x=R\\sin(x+\\alpha)$, coefficient matching gives $\\boxed{\\tan\\alpha=\\frac{p}{q}}$.","points":null,"mcqOptions":null},{"id":"cc7ea1b2-3b44-4940-8a1c-4543f0247821","cardType":"basic","front":"Why can $\\tan^{-1}\\left(\\frac{q}{p}\\right)$ give an incorrect phase angle when a coefficient is negative?","back":"It may select the wrong quadrant for the required coefficient signs.","points":2,"mcqOptions":null},{"id":"e98b5e85-941a-4237-895d-36f0ca7f7861","cardType":"basic","front":"Express $3\\cos x+4\\sin x$ as $R\\cos(x-\\alpha)$.","back":"$5\\cos\\left(x-\\tan^{-1}\\left(\\frac{4}{3}\\right)\\right)$.","points":2,"

3.4.6 Trigonometry (A-level only) Flashcards

3.4.6 Trigonometry (A-level only) Flashcards