Trigonometry Formula Sheet
Basic Trigonometric Identities
Reciprocal Identities:
sin θ=1
csc θ,csc θ=1
sin θ
cos θ=1
sec θ,sec θ=1
cos θ
tan θ=1
cot θ,cot θ=1
tan θ
Pythagorean Identities:
sin2θ+ cos2θ= 1
1 + tan2θ= sec2θ
1 + cot2θ= csc2θ
Angle Sum and Difference Formulas
For Sine:
sin(A±B) = sin Acos B±cos Asin B
For Cosine:
cos(A±B) = cos Acos B∓sin Asin B
For Tangent:
tan(A±B) = tan A±tan B
1∓tan Atan B
Double Angle Formulas
For Sine:
sin 2A= 2 sin Acos A
For Cosine:
cos 2A= cos2A−sin2A
= 2 cos2A−1
= 1 −2 sin2A
For Tangent:
tan 2A=2 tan A
1−tan2A
Half-Angle Formulas
For Sine:
sin A
2=±r1−cos A
2
For Cosine:
cos A
2=±r1 + cos A
2
For Tangent:
tan A
2=1−cos A
sin A=sin A
1 + cos A
Product to Sum Formulas
For Sine and Cosine:
sin Asin B=1
2[cos(A−B)−cos(A+B)]
cos Acos B=1
2[cos(A−B) + cos(A+B)]
sin Acos B=1
2[sin(A+B) + sin(A−B)]
Inverse Trigonometric Functions
Basic Inverse Relations:
sin−1(sin θ) = θ, cos−1(cos θ) = θ
tan−1(tan θ) = θ, cot−1(cot θ) = θ
Ranges of Inverse Functions:
sin−1x∈h−π
2,π
2i
cos−1x∈[0, π]
tan−1x∈−π
2,π
2
Law of Sines and Cosines
Law of Sines:
a
sin A=b
sin B=c
sin C
Law of Cosines:
c2=a2+b2−2ab cos C
Area of a Triangle
Using Sine:
Area = 1
2ab sin C
Using Heron’s Formula:
s=a+b+c
2
Area = ps(s−a)(s−b)(s−c)
Trigonometric Values of Special Angles
θ0◦30◦45◦60◦90◦
sin θ01
2
1
√2
√3
21
cos θ1√3
2
1
√2
1
20
tan θ01
√31√3 Undefined
