Pre-Calculus Formula Sheet
Law of Sines
𝑎
sin𝐴=𝑏
sin𝐵=𝑐
sin𝐶
Law of Cosines
𝑎2=𝑏2+𝑐2−2𝑏𝑐 cos 𝐴
𝑏2=𝑎2+𝑐2−2𝑎𝑐 cos 𝐵
𝑐2=𝑎2+𝑏2−2𝑎𝑏 cos 𝐶
Area of a Triangle
𝐴=1
2𝑎𝑏sin𝐶
𝐴=1
2𝑏𝑐sin𝐴
𝐴=1
2𝑎𝑐sin𝐵
Heron’s Formula
𝐴=√𝑠(𝑠−𝑎)(𝑠−𝑏)(𝑠−𝑐)
where 𝑠=1
2(𝑎+𝑏+𝑐)
Reciprocal Identities
sin𝑥= 1
csc𝑥
cos𝑥= 1
sec𝑥
tan𝑥= 1
cot𝑥
csc𝑥= 1
sin𝑥
sec𝑥= 1
cos𝑥
cot𝑥= 1
tan𝑥
Quotient Identities
tan𝑥=sin𝑥
cos𝑥
cot𝑥=cos𝑥
sin𝑥
Pythagorean Identities
sin2𝑥+cos2𝑥=1
tan2𝑥+1=sec2𝑥
cot2𝑥+1=csc2𝑥
Sum and Difference Identities
sin(𝐴+𝐵)=sin𝐴cos𝐵+cos𝐴sin𝐵
sin(𝐴−𝐵)=sin𝐴cos𝐵−cos𝐴sin𝐵
cos(𝐴+𝐵)=cos𝐴cos𝐵−sin𝐴sin𝐵
cos(𝐴−𝐵)=cos𝐴cos𝐵+sin𝐴sin𝐵
tan(𝐴+𝐵)= tan𝐴+tan𝐵
1−tan𝐴tan𝐵
tan(𝐴−𝐵)= tan𝐴−tan𝐵
1+tan𝐴tan𝐵
Double Angle Identities
sin(2𝑥)=2sin𝑥cos𝑥
cos(2𝑥)=cos2𝑥−sin2𝑥
=1−2sin2𝑥
=2cos2𝑥−1
tan(2𝑥)= 2tan𝑥
1−tan2𝑥
Half-Angle Identities
sin(𝑥
2)=±√1−cos𝑥
2
cos(𝑥
2)=±√1+cos𝑥
2
tan(𝑥
2)=±√1−cos𝑥
1+cos𝑥
= 1−cos𝑥
sin𝑥
= sin𝑥
1+cos𝑥
