Formula Sheet for Pre-Calculus
Pythagorean Identities
sin2θ+cos2θ= 1
1 + 𝑡𝑎𝑛2θ=𝑠𝑒𝑐2θ
1 + 𝑐𝑜𝑡2θ=𝑐𝑠𝑐2θ
Co-function identities:
sin �𝜋
2− 𝜃� =cos θ cos �𝜋
2− 𝜃� =sin θ
csc �𝜋
2− 𝜃� =sec θ tan �𝜋
2− 𝜃� =cot θ
sec �𝜋
2− 𝜃� =csc θ cot �𝜋
2− 𝜃� =tan θ
Even/odd Identities:
sin(−𝜃)=− sin θ cos(−𝜃)=cos θ
csc(−𝜃)=− csc θ tan(−𝜃)=− tan θ
sec(−𝜃)=sec θ cot(−𝜃)=− cotθ
Sum and difference formulas:
sin(𝜃±𝜑)= sin 𝜃cos 𝜑±cos 𝜃sin 𝜑
cos(𝜃±𝜑)= cos 𝜃cos 𝜑 ∓sin𝜃sin 𝜑
tan(𝜃±𝜑)= tan𝜃±tan𝜑
1∓tan𝜃tan𝜑
Half-angle formulas:
sin θ
2= ±�1−cosθ
2
cos θ
2= ±�1+cosθ
2
tan θ
2=1−cosθ
sinθ=sinθ
1+cosθ
Double-angle formulas:
sin(2𝜃)= 2 si n 𝜃cos 𝜗
cos(2𝜃)=𝑐𝑜𝑠2𝜃 −𝑠𝑖𝑛2𝜃= 2𝑐𝑜𝑠2𝜃 −1 = 1 −2𝑠𝑖𝑛2𝜃
tan(2𝜃)=2 tan𝜃
1−𝑡𝑎𝑛2𝜃
Power reducing formulas:
𝑠𝑖𝑛2𝜃=1−cos2𝜃
2
𝑐𝑜𝑠2𝜃=1+cos 2𝜃
2
𝑡𝑎𝑛2𝜃=1−cos2𝜃
1+cos2𝜃
Sum-to-product formulas:
sin 𝜃+sin 𝜑= 2 sin �𝜃+𝜑
2�cos �𝜃−𝜑
2�
sin 𝜃−sin 𝜑= 2 cos �𝜃+𝜑
2�sin �𝜃−𝜑
2�
cos 𝜃+cos 𝜑= 2 cos �𝜃+𝜑
2�cos �𝜃−𝜑
2�
cos 𝜃−cos 𝜑=−2sin �𝜃+𝜑
2�sin �𝜃−𝜑
2�
Product-to-sum formulas:
sin 𝜃sin 𝜑=1
2[cos(𝜃−𝜑)−cos(𝜃+𝜑)]
cos 𝜃cos 𝜑=1
2[cos(𝜃−𝜑)+cos(𝜃+𝜑)]
sin 𝜃cos 𝜑=1
2[sin(𝜃+𝜑)+sin(𝜃−𝜑)]
cos 𝜃sin 𝜑=1
2[sin(𝜃+𝜑)−sin(𝜃−𝜑)]
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)
