Precalculus Formulas

Here you can find formulas for a Precalculus Course.

1.1 Trigonometric functions
For the trigonometric ratios for a point p on the unit circle holds:
cos(φ) = x_p , sin(φ) = y_p , tan(φ) = y_p/x_p
sin² (x) + cos² (x) = 1 and cos⁻² (x) = 1 + tan² (x).
cos(a ± b) = cos(a) cos(b) ∓ sin(a) sin(b) ,
sin(a ± b) = sin(a) cos(b) ± cos(a)sin(b)
tan(a ± b) =( tan(a) ± tan(b) )/( 1 ∓ tan(a) tan(b) )
The sum formulas are:
sin(p) + sin(q) = 2 sin( 1 / 2 (p + q)) cos( 1 / 2 (p − q))
sin(p) − sin(q) = 2 cos( 1 / 2 (p + q)) sin( 1 / 2 (p − q))
cos(p) + cos(q) = 2 cos( 1 / 2 (p + q)) cos( 1 / 2 (p − q))
cos(p) − cos(q) = −2 sin( 1 / 2 (p + q)) sin( 1 / 2 (p − q))
From these equations can be derived that
2 cos2 (x) = 1 + cos(2x) , 2 sin2 (x) = 1 − cos(2x)
sin(π − x) = sin(x) , cos(π − x) = − cos(x)
sin( 1 / 2 π − x) = cos(x) , cos( 1 / 2 π − x) = sin(x)
Conclusions from equalities:
sin(x) = sin(a)
⇒ x = a ± 2kπ or x = (π − a) ± 2kπ, k ∈ N
cos(x) = cos(a)
⇒ x = a ± 2kπ or x = −a ± 2kπ
tan(x) = tan(a)
⇒ x = a ± kπ and x =/= π / 2 ± kπ
The following relations exist between the inverse trigonometric functions:
arctan(x) = arcsin(x/ √x² + 1)= arccos(1/√ x² + 1)

sin(arccos(x)) = √ 1 − x²