Double angle identities (A-level only) Flashcards","topicId":"38574dc0-91f5-4500-ae07-2983ca3be508","topicSlug":"double-angle-identities","cards":[{"id":"7cf90d56-44e6-4d0c-8b0a-c4861a192383","cardType":"basic","front":"

Which substitutions turn compound angle identities into double angle identities?

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Set $A=\\theta$ and $B=\\theta$.

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The sine double angle identity is $\\text{[...]}$.

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The sine double angle identity is $\\boxed{\\sin 2\\theta=2\\sin\\theta\\cos\\theta}$.

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Rewrite $\\sin\\theta\\cos\\theta$ using a double angle.

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$\\frac12\\sin 2\\theta$

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The first cosine double angle form is $\\text{[...]}$.

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The first cosine double angle form is $\\boxed{\\cos 2\\theta=\\cos^2\\theta-\\sin^2\\theta}$.

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Which form of $\\cos 2\\theta$ should be used with an expression containing only $\\cos^2\\theta$?

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$\\cos 2\\theta=2\\cos^2\\theta-1$

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Which form of $\\cos 2\\theta$ should be used with an expression containing only $\\sin^2\\theta$?

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$\\cos 2\\theta=1-2\\sin^2\\theta$

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Which form of $\\cos 2\\theta$ directly uses both $\\sin\\theta$ and $\\cos\\theta$?

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$\\cos 2\\theta=\\cos^2\\theta-\\sin^2\\theta$

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Rearranging the cosine double angle identity gives $\\sin^2\\theta=$ $\\text{[...]}$.

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Rearranging the cosine double angle identity gives $\\sin^2\\theta=$ $\\boxed{\\frac{1-\\cos 2\\theta}{2}}$.

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Rearranging the cosine double angle identity gives $\\cos^2\\theta=$ $\\text{[...]}$.

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Rearranging the cosine double angle identity gives $\\cos^2\\theta=$ $\\boxed{\\frac{1+\\cos 2\\theta}{2}}$.

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The tangent double angle identity is $\\text{[...]}$, where both sides are defined.

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The tangent double angle identity is $\\boxed{\\tan 2\\theta=\\frac{2\\tan\\theta}{1-\\tan^2\\theta}}$, where both sides are defined.

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How is the tangent double angle identity obtained from $\\frac{\\sin 2\\theta}{\\cos 2\\theta}$?

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Substitute the double angle identities, then divide by $\\cos^2\\theta$.

","points":2,"mcqOptions":null},{"id":"d2d9a78d-eb94-4d41-855e-4054f87138a1","cardType":"basic","front":"

Why is $\\tan 2\\theta$ undefined when $\\tan\\theta=\\pm1$?

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The denominator $1-\\tan^2\\theta$ equals zero.

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If $\\sin\\theta=\\frac35$ and $\\theta$ is acute, what is $\\cos\\theta$?

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$\\cos\\theta=\\frac45$

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If $\\sin\\theta=\\frac35$ and $\\theta$ is acute, find $\\sin 2\\theta$.

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$\\sin 2\\theta=\\frac{24}{25}$

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1.7.19 Double angle identities (A-level only) Flashcards

1.7.19 Double angle identities (A-level only) Flashcards