Precalculus formulas

Finding approximate point on polar graph with average rate of change formula

r(theta₂)=r(theta₁)+AROC*(theta₂-theta₁)

Polar axis symmetry formula

r(-theta)=r(theta) or symmetry about the x-axis

theta=pi/2 symmetry formula

-r(-theta)=r(theta) or symmetry about the y-axis

Pole symmetry

-r(theta)=r(theta) or symmetry about x=1

sin(theta) to cos(theta)

sin(theta)=cos(theta-pi/2)

cos(theta) to sin(theta)

cos(theta)=sin(theta+pi/2)

Odd symmetry

sin(theta)=-sin(theta) or symmetry about the origin

Even symmetry

cos(theta)=cos(-theta) or symmetry about the y-axis

csc(theta)

1/sin(theta)

sec(theta)

1/cos(theta)

cot(theta)

cos(theta)/sin(theta)

tan²(theta)

sec²theta-1

sin²+cos²=1

This fundamental identity relates the sine and cosine of an angle, stating that the square of the sine of an angle plus the square of the cosine of the same angle equals one.

sin(a+b)

sin(a)cos(b) + cos(a)sin(b)

sin(a-b)

sin(a)cos(b) - cos(a)sin(b)

cos(a+b)

cos(a)cos(b) - sin(a)sin(b)

cos(a-b)

cos(a)cos(b) + sin(a)sin(b)

sin(2a)

2sin(a)cos(a)

tan(a+b)

(tan(a)+tan(b))/(1-tan(a)tan(b))

tan(a-b)

(tan(a)-tan(b))/(1+tan(a)tan(b))

cos(2a)

cos²(a) - sin²(a)

log_b (xy)

log_b (x) + log_b (y)

log_b (x/y)

log_b (x) - log_b (y)

log_b (x)^y

y * log_b (x)

Change in base formula

log_b (x)= log(x)/log(b)

To cancel out ln

Exponentiate both sides by e

Arithmetic Sequence formula

a_n=a₁+(n-1)(d)

Arithmetic Series formula

S_n=(n/2)(2a+(n-1)(d))

Geometric Sequence formula

a_n=a₁r^(n-1)

Geometric Series formula

S_n=(a₁-(a₁rⁿ))/(1-r)

g(x)=af(x), what does the a do?

Vertical dilation by |a|

g(x)=f(bx), what does the b do?

Horizontal dilation by |1/b|

g(x)=-f(x), what does the - sign do?

Reflection over the x-axis

If the degree of the numerator > than the degree of the denominator

Then you should subtract the degrees and the ends will approach the slope of the resulting line.

If the degree of the denominator > than the degree of the numerator

The end behavior will approach the horizontal asymptote of y=0

If the degree of the numerator = the degree of the denominator

Then there will be a horizontal asymptote at the ratio of the leading coefficients.

Law of cosine

c²=a²+b²-2ab*cos(theta), helps solve missing side lengths or angles of triangles that aren’t right triangles.

Law of sine

a/sin(A) = b/sin(B) = c/sin(C), helps find missing side lengths or angles of triangles that aren’t right triangles.

Discriminant formula

The value of b²-4ac in a quadratic equation, which determines the nature of the roots.

If the discriminant is <0

There are 2 complex roots, complex roots always come in pairs and if there is a complex root a+bi, then a-bi is also going to be a root.

If the discriminant is =0

There is exactly 1 real root.

If the discriminant is a positive-perfect square

There are 2 distinct real roots, and both roots are rational.

If the discriminant is a positive - non-perfect number

There are 2 distinct real roots, but at least one root is irrational.

You only need the numerator of a rational function to find

the zeros of the function

You only need the denominator of the rational function to find

the vertical asymptotes of the function.

You only need the constants of the rational function to find

the y-intercepts of the function by dividing the two constants.

You only need the highest powered number from the numerator and denominator to find

the horizontal asymptotes of the function.

If (x,y) is on the function of f(x)

Then (y,x) is going to be on the function of f^-1(x)

The formula for finding the total amount of internal degrees for a polygon

(n-2)180

sin²(a)

(1/2)(1-cos(2a))

cos²(a)

(1/2)(1+cos(2a))

tan(2a)

(2tan(a))/(1-tan²(a))

in exponential growth, what does b in ab^x need to be

greater than 1

When you take the square root of a value

it can be positive and negative so always remember to put the plus or minus sign

the argument of a log function must always be >0

if it isn’t, it is an extraneous value

secant line

is a line that intersects a curve at two or more points, used to approximate the slope of the curve at a point.

cot

1/tan

1+tan²

sec²

cot²+1

csc²

if it doesn’t specify the domain in the question

add the period pi

concave up

rates of change are increasing

concave down

rates of change are decreasing

polar coordinates

(radius, theta)

rectangular/cartesian

(cos,sin)