All identities below hold for right triangles
x is the angle between the adjacent side and the hypotenuse.
Basic Relationships (left side = right side)
| Question (memorize) | Answer (memorize) |
|---|
| (opposite side)2 + (adjacent side)2 | (hypotenuse)2 |
| sin x | (opposite side)/hypotenuse |
| cos x | (adjacent side)/hypotenuse |
| tan x | (opposite side)/(adjacent side) |
| csc x | hypotenuse/(opposite side) |
| sec x | hypotenuse/(adjacent side) |
| cot x | (adjacent side)/(opposite side) |
| Question (memorize) | Answer (memorize) |
|---|
| sin2x + cos2x | 1 |
| tan x | (sin x)/(cos x) |
| csc x | 1/(sin x) |
| sec x | 1/(cos x) |
| cot x | (cos x)/(sin x) |
Updated: use the new 'combine' link on the right side of this page to combine tables. pools the above 2 tables
Each in terms of the others (left side = right side)
| Question (memorize) | Answer (memorize) |
|---|
| cos x | sqrt(1 - sin2x) |
| tan x | (sin x)/sqrt(1 - sin2x) |
| csc x | 1/(sin x) |
| sec x | 1/sqrt(1 - sin2x) |
| cot x | sqrt(1 - sin2x)/(sin x) |
| Question (memorize) | Answer (memorize) |
|---|
| sin x | sqrt(1 - cos2x) |
| tan x | sqrt(1 - cos2x)/(cos x) |
| csc x | 1/sqrt(1 - cos2x) |
| sec x | 1/(cos x) |
| cot x | (cos x)/sqrt(1 - cos2x) |
| Question (memorize) | Answer (memorize) |
|---|
| sin x | (tan x)/sqrt(1 + tan2x) |
| cos x | 1/sqrt(1 + tan2x) |
| csc x | sqrt(1 + tan2x)/(tan x) |
| sec x | sqrt(1 + tan2x) |
| cot x | 1/(tan x) |
| Question (memorize) | Answer (memorize) |
|---|
| sin x | 1/(csc x) |
| cos x | sqrt(csc2 - 1)/(csc x) |
| tan x | 1/sqrt(csc2x - 1) |
| sec x | (csc x)/sqrt(csc2x - 1) |
| cot x | sqrt(csc2x - 1) |
| Question (memorize) | Answer (memorize) |
|---|
| sin x | sqrt(sec2x - 1)/(sec x) |
| cos x | 1/(sec x) |
| tan x | sqrt(sec2x - 1) |
| csc x | (sec x)/sqrt(sec2x - 1) |
| cot x | 1/sqrt(sec2x - 1) |
| Question (memorize) | Answer (memorize) |
|---|
| sin x | 1/sqrt(1 + cot2x) |
| cos x | (cot x)/sqrt(1 + cot2x) |
| tan x | 1/(cot x) |
| csc x | sqrt(1 + cot2x) |
| sec x | sqrt(1 + cot2x)/(cot x) |
Updated: use the new 'combine' link on the right side of this page to combine tables. pools the last 6 tables
Double-Angle Formulas (left side = right side)
| Question (memorize) | Answer (memorize) |
|---|
| sin(2x) | 2 sin x cos x |
| sin(2x) | (2 tan x)/(1 + tan2x) |
| cos(2x) | cos2x - sin2x |
| cos(2x) | 2 cos2 x - 1 |
| cos(2x) | 1 - 2 sin2x |
| cos(2x) | (1 - tan2x)/(1 + tan2x) |
| tan(2x) | (2 tan x)/(1 - tan2x) |
| cot(2x) | (cot2x - 1)/(2 cot x) |
Triple-Angle Formulas (left side = right side)
| Question (memorize) | Answer (memorize) |
|---|
| sin(3x) | 3 sin x - 4 sin3x |
| cos(3x) | 4 cos3x - 3 cos x |
| tan(3x) | (3 tan x - tan3x)/(1 - 3 tan2x) |
| cot(3x) | (3 cot x - cot3x)/(1 - 3 cot2x) |
Half-Angle Formulas (left side = right side)
| Question (memorize) | Answer (memorize) |
|---|
| sin(x/2) | ±sqrt[(1 - cos x)/2] |
| cos(x/2) | ±sqrt[(1 + cos x)/2] |
| tan(x/2) | csc x - cot x |
| tan(x/2) | ±sqrt[(1 - cos x)/(1 + cos x)] |
| tan(x/2) | (sin x)/(1 + cos x) |
| tan(x/2) | (1 - cos x)/(sin x) |
| cot(x/2) | csc x + cot x |
| cot(x/2) | ±sqrt[(1 + cos x)/(1 - cos x)] |
| cot(x/2) | (sin x)/(1 - cos x) |
| cot(x/2) | (1 + cos x)/(sin x) |
Updated: use the new 'combine' link on the right side of this page to combine tables. pools the last 3 tables
Angle-Addition Formulas (left side = right side)
| Question (memorize) | Answer (memorize) |
|---|
| sin(x+y) | sin x cos y + cos x sin y |
| sin(x-y) | sin x cos y - cos x sin y |
| cos(x+y) | cos x cos y - sin x sin y |
| cos(x-y) | cos x cos y + sin x sin y |
| tan(x+y) | (tan x + tan y)/(1 - tan x tan y) |
| tan(x-y) | (tan x - tan y)/(1 + tan x tan y) |
| tan[(x+y)/2] | (sin x + sin y)/(cos x + cos y) |
| tan[(x+y)/2] | -(cos x - cos y)/(sin x - sin y) |
Product-to-Sum Formulas (left side = right side)
| Question (memorize) | Answer (memorize) |
|---|
| cos x cos y | [cos(x-y) + cos(x+y)]/2 |
| sin x sin y | [cos(x-y) - cos(x+y)]/2 |
| sin x cos y | [sin(x+y) + sin(x-y)]/2 |
| cos x sin y | [sin(x+y) - sin(x-y)]/2 |
Sum-to-Product Formulas (left side = right side)
| Question (memorize) | Answer (memorize) |
|---|
| sin x + sin y | 2 sin[(x+y)/2] cos[(x-y)/2] |
| cos x + cos y | 2 cos[(x+y)/2] cos[(x-y)/2] |
| cos x - cos y | -2 sin[(x+y)/2] sin[(x-y)/2] |
| sin x - sin y | 2 cos[(x+y)/2] sin[(x-y)/2] |
Updated: use the new 'combine' link on the right side of this page to combine tables. pools the last 3 tables
Miscellaneous Formulas (left side = right side)
| Question (memorize) | Answer (memorize) |
|---|
| cos x + i sin x | eix |
| cos(-x) + i sin(-x) | e-ix |
| cos x - i sin x | e-ix |
| -1 | i2 |
| -1 | eiπ |
| sin x | (eix - e-ix)/(2i) |
| cos x | (eix + e-ix)/(2) |
| tan x | (eix - e-ix)/[i(eix + e-ix)] |
| cos(nx) + i sin(nx) | (cos x + i sin x)n |
| tan[(n+1)x] | [tan(nx) + tan(x)]/[1 - tan(nx) tan(x)] |
| cot[(n+1)x] | [cot(nx) cot(x) - 1]/[cot(nx) + cot(x)] |
Reference: http://en.wikipedia.org/wiki/Trigonometric_identity
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