Solving Trig Equations

When solving a trig equation, these guidelines may be helpful:
*you may want to sketch a quick graph to see what the solutions are
*transform the functions of 2x into functions of x by using identities, if the equation involves functions of 2x and x
*if the equation involves functions of only 2x, you would probably want to solve for 2x first and then x afterwards
*make sure you don't lose roots when dividing both sides by a function of a variable

Example 1:
sec^2theta=9
square root both sides
sec(theta)=+/-3degrees
theta=sec^-1(3)
theta=cos^-1(1/3)
=70.529
you have to take the value of all four quadrants because you get a positive and negative number when you square root
theta=70degrees31'44''
109degrees28'15''
250degrees31'44''
289degrees28'15''

Example 2:
tan^2theta=1
square root both sides
tan(theta)=+/-1
theta=tan^-1(1)
tan is positive and negative in all four quadrants
theta=45degrees
135degrees
225degrees
315degrees

Example 3:
1-csc^2(theta)=-3
subtract 1 from both sides
-csc^2(theta)=-4
divide both side by -1
csc^2(theta)=4
square root both sides
csc(theta)=+/-2
theta=csc^-1(2)
sin is positive and negative in all four quadrants
theta=30degrees
150degrees
210degrees
330degrees

Example 4:
8cos^2(theta)-3=1
add 3 to both sides
8cos^2(theta)=4
divide both sides by 8
cos^2(theta)=1/2
square root both sides
theta=cos^-1(square root of 2/2
theta=45degrees
135degrees
225degrees
315degrees

Example 5:
6sin^2(theta)-7sin(theta)+2=0
factor
(6sin^2-4sin)-(3sin+2)=0
group and factor again
2sin(3sin-2)-1(3sin-2)=0
set both equal to 0
2sinx-1=0 & 3sinx-2=0
sinx=1/2 sinx=2/3
x=sin^-1(1/2) x=sin^-1(2/3)
sine is positive in the first and second quadrants
x=30degrees,150degrees x=41.810degrees,138.19degrees
x=30degrees,41degrees48'36'',138degrees11'24'',150degrees