Question 956325: Algebraically determine the solution to the following equations. Provide answers that are appropriate taking into consideration the stated domains.
 over the domain, -180 degrees <(or equal to) x <(or equal to) 180 degrees
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! start with 7 * cos(theta) - 2 = -3
add 2 to both sides of the equation to get:
7 * cos(theta) = -1
divide both sides of the equation by 7 to get:
cos(theta) = -1/7
to find the reference angle, make cos(theta) = positive.
you will get cos(theta) = 1/7
use your calculator to solve for theta to get:
theta = arccos(1/7) = 81.7867893 degrees.
since cos(theta) needs to be negative, then your angle has to be in quadrant 2 or quadrant 3.
your angle in quadrant 2 is equal to 180 - 81.7867893 which is equal to 98.2132107 degrees.
your angle in quadrant 3 is equal to 180 + 81.7867893 which is equal to 261.7867893 degrees.
your interval is from -180 to 180.
261.7867893 degrees doesn't work because it is greater than 180.
since trig functions are repetitive, then your solution is actually:
98.2132107 plus or minus k * 360, where k is a non-negative integer, ....
and 261.7867893 plus or minus k * 360, where k is a non-negative integer.
98.2132107 is within your domain of -180 to 180, so that solution is good.
261.7867893 - 1 * 360 is equal to -98.2132107 which is also within your domain, so that solution is also good.
none of the other solutions are within your domain, so those two solutions are the only valid solutions that are within the required domain interval.
they are:
theta = 98.2132107 degrees and theta = -98.2132107 degrees.
a graph of your equation is shown below:
this graph shows both equations separately.
the solution is the intersection of the graphs of the two equations.
first equation is y = 7 * cos(theta) - 2
second equation is y = -3
the intersection of the two graphs is when y = 7 * cos(theta) - 1 is equal to y = -3.
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