Lesson Compendium of Trigonometry formulas

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Compendium of Trigonometry formulas


This lesson is the collection of formulas expressing properties of Trigonometric Functions.

Simplest relations


sin%28alpha%29%5E2+%2B+cos%28alpha%29%5E2+=+1,
sin%28-alpha%29+=+-sin%28alpha%29, cos%28-alpha%29+=+cos%28alpha%29,

tan%28alpha%29+=+sin%28alpha%29%2Fcos%28alpha%29, cot%28alpha%29+=+cos%28alpha%29%2Fsin%28alpha%29,
tan%28-alpha%29+=+-tan%28alpha%29, cot%28-alpha%29+=+-cot%28alpha%29,

1+%2B+tan%5E2%28alpha%29+=+1%2Fcos%5E2%28alpha%29, 1+%2B+cot%5E2%28alpha%29+=+1%2Fsin%5E2%28alpha%29.

Reduction formulas


sin%28alpha+%2B-+2%2Api%2An%29+=+sin%28alpha%29, cos%28alpha+%2B-+2%2Api%2An%29+=+cos%28alpha%29,

sin%28alpha+%2B-+pi%2An%29+=+%28-1%29%5En%2Asin%28alpha%29, cos%28alpha+%2B-+pi%2An%29+=+%28-1%29%5En%2Acos%28alpha%29,

sin%28alpha+%2B+pi%2F4%29+=+%28sqrt%282%29%2F2%29%2A%28sin%28alpha%29+%2B+cos%28alpha%29%29, cos%28alpha+%2B+pi%2F4%29+=+%28sqrt%282%29%2F2%29%2A%28cos%28alpha%29-+sin%28alpha%29%29,

sin%28alpha+-+pi%2F4%29+=+%28sqrt%282%29%2F2%29%2A%28sin%28alpha%29+-+cos%28alpha%29%29, cos%28alpha+-+pi%2F4%29+=+%28sqrt%282%29%2F2%29%2A%28cos%28alpha%29%2B+sin%28alpha%29%29,

tan%28alpha+%2B-+pi%2An%29+=+tan%28alpha%29, cot%28alpha+%2B-+pi%2An%29+=+cot%28alpha%29.

For the proof and examples see the lesson The Amazing Unit Circle: Trigonometric Identities in this module.

Addition and subtraction formulas


cos%28alpha+%2B+beta%29+=+cos%28alpha%29%2Acos%28beta%29+-+sin%28alpha%29%2Asin%28beta%29,
cos%28alpha+-+beta%29+=+cos%28alpha%29%2Acos%28beta%29+%2B+sin%28alpha%29%2Asin%28beta%29,

sin%28alpha+%2B+beta%29+=+sin%28alpha%29%2Acos%28beta%29+%2B+cos%28alpha%29%2Asin%28beta%29,
sin%28alpha+-+beta%29+=+sin%28alpha%29%2Acos%28beta%29+-+cos%28alpha%29%2Asin%28beta%29,

tan%28alpha+%2B+beta%29+=+%28tan%28alpha%29+%2B+tan%28beta%29%29%2F%281+-+tan%28alpha%29%2Atan%28beta%29%29,

tan%28alpha+-+beta%29+=+%28tan%28alpha%29+-+tan%28beta%29%29%2F%281+%2B+tan%28alpha%29%2Atan%28beta%29%29.

For the proof and examples see the lessons Addition and subtraction formulas and
Addition and subtraction formulas - Examples in this module.

Addition and subtraction of trigonometric functions


sin%28alpha%29+%2B+sin%28beta%29+=+2%2Asin%28%28alpha%2Bbeta%29%2F2%29%2Acos%28%28alpha-beta%29%2F2%29,

sin%28alpha%29+-+sin%28beta%29+=+2%2Asin%28%28alpha-beta%29%2F2%29%2Acos%28%28alpha%2Bbeta%29%2F2%29,

cos%28alpha%29+%2B+cos%28beta%29+=+2%2Acos%28%28alpha%2Bbeta%29%2F2%29%2Acos%28%28alpha-beta%29%2F2%29,

cos%28alpha%29+-+cos%28beta%29+=+-2%2Asin%28%28alpha%2Bbeta%29%2F2%29%2Asin%28%28alpha-beta%29%2F2%29,


tan%28alpha%29+%2B-+tan%28beta%29+=+sin%28alpha+%2B-+beta%29%2F%28cos%28alpha%29%2Acos%28beta%29%29,

cot%28alpha%29+%2B-+cot%28beta%29+=+sin%28alpha+%2B-+beta%29%2F%28sin%28alpha%29%2Asin%28beta%29%29.

For the proof and examples see the lessons Addition and subtraction of trigonometric functions and
Addition and subtraction of trigonometric functions - Examples in this module.

Product of trigonometric functions


sin%28alpha%29%2Asin%28beta%29+=+%281%2F2%29%2A%28cos%28alpha-beta%29+-+cos%28alpha%2Bbeta%29%29,

cos%28alpha%29%2Acos%28beta%29+=+%281%2F2%29%2A%28cos%28alpha-beta%29+%2B+cos%28alpha%2Bbeta%29%29,

sin%28alpha%29%2Acos%28beta%29+=+%281%2F2%29%2A%28sin%28alpha-beta%29+%2B+sin%28alpha%2Bbeta%29%29.

For the proof and examples see the lessons Product of trigonometric functions and
Product of trigonometric functions - Examples in this module.

Powers of trigonometric functions


cos%5E2%28alpha%29+=+%281%2F2%29%2Acos%282alpha%29+%2B+1%2F2,

sin%5E2%28alpha%29+=+-%281%2F2%29%2Acos%282alpha%29+%2B+1%2F2,

cos%5E3%28alpha%29+=+%281%2F4%29%2Acos%283alpha%29+%2B+%283%2F4%29%2Acos%28alpha%29,

sin%5E3%28alpha%29+=+-%281%2F4%29%2Asin%283alpha%29+%2B+%283%2F4%29%2Asin%28alpha%29.

For the proof and examples see the lessons Powers of Trigonometric functions and
Powers of Trigonometric functions - Examples in this module.

Trigonometric functions of multiply argument


cos%282alpha%29+=+2%2Acos%5E2%28alpha%29+-+1,

sin%282alpha%29+=+2%2Asin%28alpha%29%2Acos%28alpha%29,

cos%283alpha%29+=+4%2Acos%5E3%28alpha%29+-+3%2Acos%28alpha%29,

sin%283alpha%29+=+-4%2Asin%5E3%28alpha%29+%2B+3%2Asin%28alpha%29.

For the proof and examples see the lessons Trigonometric functions of multiply argument and
Trigonometric functions of multiply argument - Examples in this module.

Trigonometric functions of half argument


sin%5E2%28alpha%2F2%29+=+%281-cos%28alpha%29%29%2F2, cos%5E2%28alpha%2F2%29+=+%281%2Bcos%28alpha%29%29%2F2,

tan%28alpha%2F2%29+=+sin%28alpha%29%2F%281%2Bcos%28alpha%29%29+=+%281-cos%28alpha%29%29%2Fsin%28alpha%29,

sin%28alpha%29+=+2%2Atan%28alpha%2F2%29%2F%281%2Btan%5E2%28alpha%2F2%29%29, cos%28alpha%29+=+%281-tan%5E2%28alpha%2F2%29%29%2F%281%2Btan%5E2%28alpha%2F2%29%29, tan%28alpha%29+=+2%2Atan%28alpha%2F2%29%2F%281-tan%5E2%28alpha%2F2%29%29.

For the proof and examples see the lessons Trigonometric functions of half argument and
Trigonometric functions of half argument - Examples in this module.
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