Proofs with trigonometric identities Flashcards","topicId":"de74cbad-b351-4549-a303-5b583fd28577","topicSlug":"proofs-with-trigonometric-identities","cards":[{"id":"fcb5ae15-0d9d-408d-9efb-04811034d0b7","cardType":"cloze","front":"

The fundamental Pythagorean identity is $\\text{[...]}$.

","back":"

The fundamental Pythagorean identity is $\\boxed{\\sin^2x+\\cos^2x\\equiv1}$.

","points":null,"mcqOptions":null},{"id":"89a56002-67c1-4cf2-97fc-92e5efe31dd7","cardType":"cloze","front":"

The quotient identity is $\\text{[...]}$.

","back":"

The quotient identity is $\\boxed{\\tan x\\equiv\\frac{\\sin x}{\\cos x}}$.

","points":null,"mcqOptions":null},{"id":"89b9d28a-002b-4fd5-8209-2806c655f887","cardType":"cloze","front":"

The reciprocal identity for secant is $\\text{[...]}$.

","back":"

The reciprocal identity for secant is $\\boxed{\\sec x\\equiv\\frac{1}{\\cos x}}$.

","points":null,"mcqOptions":null},{"id":"9c7971f9-458d-4076-b4aa-fb54bc198124","cardType":"cloze","front":"

The reciprocal identity for cosecant is $\\text{[...]}$.

","back":"

The reciprocal identity for cosecant is $\\boxed{\\cosec x\\equiv\\frac{1}{\\sin x}}$.

","points":null,"mcqOptions":null},{"id":"2f7479e4-a449-4e1a-88c2-5af1b4c769c3","cardType":"cloze","front":"

Dividing the Pythagorean identity by $\\cos^2x$ gives $\\text{[...]}$.

","back":"

Dividing the Pythagorean identity by $\\cos^2x$ gives $\\boxed{1+\\tan^2x\\equiv\\sec^2x}$.

","points":null,"mcqOptions":null},{"id":"c2ca1ff0-d3e3-41ea-a1fb-1c02941704a2","cardType":"cloze","front":"

Dividing the Pythagorean identity by $\\sin^2x$ gives $\\text{[...]}$.

","back":"

Dividing the Pythagorean identity by $\\sin^2x$ gives $\\boxed{1+\\cot^2x\\equiv\\cosec^2x}$.

","points":null,"mcqOptions":null},{"id":"d303c54f-d4be-4c9a-8c46-d84568657bec","cardType":"basic","front":"

Why is $\\sin x=0$ an equation rather than an identity?

","back":"

It is true only for particular values of $x$.

","points":2,"mcqOptions":null},{"id":"6e073d09-b7d1-49d9-b6f9-87c2ec76c536","cardType":"basic","front":"

Why do several successful numerical substitutions not prove an identity?

","back":"

They do not establish equality for all permitted values.

","points":2,"mcqOptions":null},{"id":"7f8d4d3a-6110-431c-a2b9-c7b80b5c71cf","cardType":"basic","front":"

Which side should you normally simplify when proving an identity?

","back":"

Start with the more complicated side.

","points":1,"mcqOptions":null},{"id":"4d2cce86-e3bd-4753-9c16-31dd812827a3","cardType":"basic","front":"

Why is altering both sides simultaneously risky in an identity proof?

","back":"

It can introduce circular reasoning by assuming the result.

","points":2,"mcqOptions":null},{"id":"869cbad9-e845-416b-b591-14a4f3359ec3","cardType":"basic","front":"

What does $1-\\sin^2x$ simplify to?

","back":"

$\\cos^2x$

","points":1,"mcqOptions":null},{"id":"879ba049-204b-4419-b3a7-8e675110e3c7","cardType":"basic","front":"

What does $1-\\cos^2x$ simplify to?

","back":"

$\\sin^2x$

","points":1,"mcqOptions":null},{"id":"dbd5eeb0-2777-4065-b227-2f0d59b03394","cardType":"basic","front":"

What may be cancelled in a trigonometric fraction?

","back":"

Only a common factor, not separate terms.

","points":1,"mcqOptions":null},{"id":"5daaf711-25ab-490c-ac62-8246f052d2f8","cardType":"basic","front":"

What happens to an original domain restriction after its factor is cancelled?

","back":"

The original restriction still applies.

","points":1,"mcqOptions":null},{"id":"3b271eb8-4960-41f1-8970-237a28a5bddb","cardType":"basic","front":"

Why can rewriting $\\tan x$, $\\sec x$, $\\cosec x$ and $\\cot x$ using sine and cosine hel

1.8.8 Proofs with trigonometric identities Flashcards

1.8.8 Proofs with trigonometric identities Flashcards