Updated: October 2019
Reciprocal Identities:
sin 𝜃𝜃=1
csc 𝜃𝜃
csc 𝜃𝜃=1
sin 𝜃𝜃
cos 𝜃𝜃=1
sec 𝜃𝜃
sec 𝜃𝜃=1
cos 𝜃𝜃
tan 𝜃𝜃=1
cot 𝜃𝜃
cot 𝜃𝜃=1
tan 𝜃𝜃
Trigonometry Formulas and Properties
Tangent and Cotangent Identities:
tan 𝜃𝜃=sin 𝜃𝜃
cos 𝜃𝜃
cot =cos 𝜃𝜃
sin 𝜃𝜃
Pythagorean Identities:
sin2𝜃𝜃+cos2𝜃𝜃= 1
tan2𝜃𝜃+ 1 = sec2𝜃𝜃
1 + cot2𝜃𝜃=csc2𝜃𝜃
Even/Odd Formulas:
sin(−𝜃𝜃)=−sin 𝜃𝜃
cos(−𝜃𝜃)=cos 𝜃𝜃
tan(−𝜃𝜃)=−tan 𝜃𝜃
csc(−𝜃𝜃)=−csc 𝜃𝜃
sec(−𝜃𝜃)=sec 𝜃𝜃
cot(−𝜃𝜃)=−cot 𝜃𝜃
Cofunction Formulas:
sin �𝜋𝜋
2−𝜃𝜃�=cos 𝜃𝜃
csc �𝜋𝜋
2−𝜃𝜃�=sec 𝜃𝜃
tan �𝜋𝜋
2−𝜃𝜃�=cot 𝜃𝜃
cos �𝜋𝜋
2−𝜃𝜃�=sin 𝜃𝜃
sec �𝜋𝜋
2−𝜃𝜃�=csc 𝜃𝜃
cot �𝜋𝜋
2−𝜃𝜃�=tan 𝜃𝜃
Product to Sum Formulas:
sin 𝛼𝛼sin 𝛽𝛽=1
2[cos(𝛼𝛼−𝛽𝛽)−cos(𝛼𝛼+𝛽𝛽)]
cos 𝛼𝛼cos 𝛽𝛽=1
2[cos(𝛼𝛼−𝛽𝛽)+cos(𝛼𝛼+𝛽𝛽)]
sin 𝛼𝛼cos 𝛽𝛽=1
2[sin(𝛼𝛼+𝛽𝛽)+sin(𝛼𝛼−𝛽𝛽)]
cos 𝛼𝛼sin 𝛽𝛽=1
2[sin(𝛼𝛼+𝛽𝛽)−sin(𝛼𝛼−𝛽𝛽)]
Sum to Product Formulas:
sin 𝛼𝛼+sin 𝛽𝛽= 2 sin �𝛼𝛼+𝛽𝛽
2�cos �𝛼𝛼−𝛽𝛽
2�
sin 𝛼𝛼−sin 𝛽𝛽= 2 cos �𝛼𝛼+𝛽𝛽
2�sin �𝛼𝛼−𝛽𝛽
2�
cos 𝛼𝛼+cos 𝛽𝛽= 2 cos �𝛼𝛼+𝛽𝛽
2�cos �𝛼𝛼−𝛽𝛽
2�
cos 𝛼𝛼−cos 𝛽𝛽=−2sin �𝛼𝛼+𝛽𝛽
2�sin �𝛼𝛼−𝛽𝛽
2�
Sum and Difference Formulas
:
sin(𝛼𝛼±𝛽𝛽) = sin𝛼𝛼cos 𝛽𝛽±sin 𝛽𝛽cos 𝛼𝛼
cos(𝛼𝛼±𝛽𝛽) = cos𝛼𝛼cos 𝛽𝛽∓sin 𝛼𝛼sin 𝛽𝛽
tan(𝛼𝛼±𝛽𝛽) = tan 𝛼𝛼±tan 𝛽𝛽
1∓tan 𝛼𝛼tan 𝛽𝛽
Periodic Formulas
:
sin(𝜃𝜃+ 2𝜋𝜋𝜋𝜋)=sin 𝜃𝜃
csc(𝜃𝜃+ 2𝜋𝜋𝜋𝜋)=csc 𝜃𝜃
cos(𝜃𝜃+ 2𝜋𝜋𝜋𝜋)=cos 𝜃𝜃
sec(𝜃𝜃+ 2𝜋𝜋𝜋𝜋)=sec 𝜃𝜃
tan(𝜃𝜃+𝜋𝜋𝜋𝜋)=tan 𝜃𝜃
cot(𝜃𝜃+𝜋𝜋𝜋𝜋)=cot 𝜃𝜃
Half-Angle Formulas:
sin �𝜃𝜃
2�= ±�1−cos 𝜃𝜃
2
cos �𝜃𝜃
2�= ±�1 + cos 𝜃𝜃
2
tan �𝜃𝜃
2�= ±�1−cos 𝜃𝜃
1 + cos 𝜃𝜃
Double Angle Formulas:
sin(2𝜃𝜃)= 2 sin 𝜃𝜃cos 𝜃𝜃
cos(2𝜃𝜃)=cos2𝜃𝜃−sin2𝜃𝜃
= 2 cos
2
𝜃𝜃−1
= 1 −2sin2𝜃𝜃
tan 2𝜃𝜃=2tan 𝜃𝜃
1−tan2𝜃𝜃
