pre calculus urgent help with tan, sec, csc, cot?

Question

how do i determine the period, vertical shift, phase shift, domain, range, and asymptote of these problems:

*by pi i mean 3.14...... the n with the swigly line at the top i dunno how to type it on the keyboard

1. y = 1 - tan ( x/2 - pi/8 )

2. y = 3 + 2 csc ( x/2 - pi/3 )

3. y = 1 + cot ( 3x + pi/2 )

4. y = 2 - sec ( 2x - pi/2 )

Answer 1

All of these are of the basic form:

y = a + b sin (k x + c)

In this formula (which works for all trig functions sin, cos, tan, cot, sec, csc) the following is true:

a is the vertical shift
b is the amplitude
c is the phase shift
k is the angular frequency

To get period we need to find the reciprocal of frequency, which can be determined by using the following formula:

f = k / (2 pi)

To find period:

T = 1 / f

Domain refers to the range of continuous x-values for which you can obtain a solution. Range refers to the range of possible solutions (y-values) to the function. Asymptote is a point where the function has no solution.

Answer 2

tan = y/x => reverse aspect = y and adjoining aspect = x hypotenuse = ?(x^two + y^two) sin = y/?(x^two + y^two) and csc = ?(x^two + y^two) / y cos = x /?(x^two + y^two) and sec = ?(x^two + y^two) /x cot = one million/tan = x /y