pre calc equations 3

pythagorean equation

quadratic formula

reference angle

amplitude and period: y= a sin/cos (bx)

amp= a, period= 2π/b

amplitude and period: tan/ cot

period= π/b

reciprocal identities

sinx= 1/cscx, cosx= 1/secx, tan= 1/cotx

ratio identities

tanx= sinx/cosx, cotx= cox/sinx

pythagorean trig identities

double-angle identity for sine

sin (2θ)= 2sin(θ)cos(θ)

double-angle identity for cosine

double-angle identity for tangent

power-reducing trig identity

half-angle identities: tangent, sine & cosine

definition, domain, range: y=sin^-1(x) (or y=arcsin (x))

• y=sin^-1(x) if and only if x=sin(y)

• domain: [-1,1]

• range: [-π/2, π/2]

• So, y=sin^-1(x) lies in the 1st or 4th quadrant

definition, domain, range: y=cos^-1(x) (or y=arccos (x))

• y = cos^-1(x) if and only if x=cos(y)

• domain: [-1,1]

• range: [0, π]

• So, y=cos^-1(x) lies in the 1st or 2nd quadrant

definition, domain, range: y=tan^-1(x) (or y=arctan (x))

• y = tan^-1(x) if and only if x=tan(y)

• domain: (-∞,∞)

• range: (-π/2, π/2)

• So, y=tan^-1(x) lies in the 1st or 4nd quadrant

the law of sines

if A,B, and C are the measures of the angles of a triangle, and a, b, and c are the lengths of the sides opposite these angles, then:

a/sin(A) = b/sin(B) = c/sin(C)

the law of cosines

area of a triangle (SAS form)

• K=1/2bc sin(A)

• K=1/2ac sin(B)

• K=1/2ab sin(C)

heron’s formula for the area of a triangle (SSS form)

If a,b and c are the lengths of the sides of a triangle, then the area of the triangle is

• √s(s-a)(s-b)(s-c)

where s=1/2(a+b+c) is the semiperimeter

definition of an inverse function

obtained by switching its input and output values ( results in the flip-flopping of the domain and range)

horizontal line test

the function is 1 to 1 if no horizontal line intersects its graph at more than one point

how to show 2 functions f and g are inverses

show that

• f(g(x))=x

and

• g(f(x))=x

steps for finding the inverse of a function f(x)

1. Verify that f is 1 to t (HLT)

2. Substitute y for f(x)

3. Switch x and y

4. Solve for y

5. Substitute f^-1(x) for y

(restrict the domain of f^-1 to match the range of f if necessary)

definition of an exponential function with base a

f(x)= a^x, where a>0, a doesn’t equal zero, and x is any reak number

the definition of a logarithmic function with base a

y=loga(x) is equal to a^y=x for x>0, a>0, and a doesn’t equal 1

exponential to log