Trigonometry Cheatsheet

Trigonometry Cheatsheet
Version 2.1
© 2008 Ankur Banerjee
* sin (A+B)
* cos (A+B)
=
=
sinA cosB + cosA sinB
cosA cosB – sinA sinB
2
2
2
2
=
=
=
=
cos (A+B) + cos (A–B)
cos (A–B) – cos (A+B)
sin (A+B) + sin (A–B)
sin (A+B) – sin (A–B)
cosA cosB
sinA sinB
sinA cosB
cosA sinB
sin(A+B) sin(A–B)
cos(A+B) cos(A–B)
=
=
sin2A – sin2B
cos2A – sin2B
sinA + sinB
sinA – sinB
cosA + cosB
cosA – cosB
=
=
=
=
2 sin[(A+B) / 2] cos[(A–B) / 2]
2 cos[(A+B) / 2] sin[(A–B) / 2]
2 cos[(A+B) / 2] cos[(A–B) / 2]
–2 sin[(A+B) / 2] sin[(A–B) / 2]
* tan (A+B)
=
tanA + tanB
1 – tanA tanB
* cot (A+B)
=
cotA cotB – 1
cotB + cotA
sin 2A
=
2 sinA cosA
cos 2A
=
cos2A – sin2A =
tan 2A
=
2 tanA
1 – tan2A
2 sin2A
2 cos2A
=
=
1 – cos2A
1 + cos2A
=
2 tanA
1 + tan2A
1 – tan2A
1 + tan2A
* Substitute '+' with '−' to get corresponding formula
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tan2(A/2) =
1 – cosA
1 + cosA
sin 3A
=
=
3 sinA – 4 sin3A
4 sinA sin(60° – A) sin(60° + A)
cos 3A
=
=
4 cos3A – 3 cosA
4 cosA cos(60° – A) cos(60° + A)
tan 3A
=
3 tanA – tan3A
1 – 3 tan2A
tanA tan(60° – A) tan(60° + A)
=
General Solutions
For sinA = k; A = nπ + (–1)n B
For cosA = k; A = 2nπ ± B
For tanA= k; A = nπ + B
For sin2A = sin2B, cos2A = cos2B, tan2A = tan2B;
A = nπ ± B
sinA + sin(A+B) + sin(A+2B) + ... + sin(A + (n–1)B)
= [ sin(A + (n–1)B/2) sin(nB/2) ] / sin(B/2)
cosA + cos(A+B) + cos(A+2B) + ... + cos(A + (n–1)B)
= [ cos(A + (n–1)B/2) sin(nB/2) ] / sin(B/2)
In a triangle ABC
∑ sin A
=
4 ∏ cos(A/2)
∑ cos A
=
1 + 4 ∏ sin(A/2)
∑ sin 2A
=
4 ∏ sinA
∑ cos 2A
=
–1 – 4∏ cosA
∑ tan A
=
∏ tanA
∑ tan(A/2) tan(B/2)
=
1
∑ cotA cotB
=
1
∑ cosA + cos(∑A) =
4 ∏ cos[(A+B)/2]
∑ sinA – sin(∑A)
=
4 ∏ sin[(A+B)/2 ]
tan (∑A)
=
∑ tanA – ∏ tanA
1 – ∑ tanA tanB
Sine Rule:
Cosine Rule:
Projection Rule:
(sinA / a) = (sinB / b) = (sinC / c)
a2 = b2 + c2 + 2 bc cosA
a = b cosC + c cosB
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