Question 1161283: I am having a lot of difficulty with one of my practice questions for Trigonometry Identity, any guidance/ help given is greatly appreciated!
Solve the following equation: 2 cos 3x+ cos 2x+1=0.
Thank you!!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! her's a list of trigonometric identities that might help.
https://en.wikipedia.org/wiki/List_of_trigonometric_identities
that's the first part of the battle.
your equation is:
2 * cos(3x) + cos(2x) + 1 = 0
from the list of identities you get:
cos(3x) = 4 * cos^3(x) - 3 * cos(x)
cos(2x) = cos^2(x) - sin^2(x)
replace cos(3x) with it's identity and replace cos(2x) with its identity to get:
2 * (4 * cos^3(x) - 3 * cos(x)) + cos^2(x) - sin^2(x) + 1 = 0
simplify to get:
8 * cos^3(x) - 6 * cos(x) + cos^2(x) - sin^2(x) + 1 = 0
since sin^(x) = 1 - cos^2(x), the equation becomes:
8 * cos^3(x) - 6 * cos(x) + cos^2(x) - (1 - cos^2(x)) + 1 = 0
simplify to get:
8 * cos^3(x) - 6 * cos(x) + cos^2(x) - 1 + cos^2(x) + 1 = 0
combine like terms to get:
8 * cos^3(x) - 6 * cos(x) + 2 * cos^2(x) = 0
divide both sides of the equation by cos(x0 to get:
8 * cos^2(x) - 6 + 2 * cos(x) = 0
divide both sides of the equation by 2 and order the terms in descending order of degree to get:
4 * cos^2(x) + cos(x) - 3 = 0
factor this quadratic equation to get:
4 * cos(x) - 3) * (cos(x) + 1) = 0
solve for cos(x) to get:
cos(x) = 3/4 or cos(x) = -1
solve for x to get:
x = 180 degrees or x = 41.40962211 degrees.
confirm your solution is correct by replacing x with those values in the original equation.
the original equation is:
2 * cos(3x) + cos(2x) + 1 = 0
when x = 180 degrees, the equation becomes:
2 * cos(540) + cos(360) + 1 = 0
evaluate to get 0 = 0, confirming that 180 degrees is a solution.
when x = 41.40962211 degrees, the equation becomes:
2 * cos(3 * 41.40962211) + cos(41.40962211) + 1 = 0
evaluate to get 0 = 0, confirming that 41.40962211 degrees is also a solution.
here's what the graph of that equation looks like.
the graph confirms that x = 41.... and x = 180 are correct.
the graph also shows there are other angles that satisfy the equation.
these would be equivalent angles, i.e. angles in other quadrants that also satisfy the equation.
not sure if you need to know this, but i'll go through one of the other angles to show you that the equation is true when you get those other angles.
for example:
x = 540 is one of those angles.
the equation is 2 * cos(3x) + cos(2x) + 1 = 0
replace x with 540 and evaluate the equation to get 0 = 0, confirming that angle also satisfies the equation.
another angle is 318.59.
that would be 41.40962211) equivalent angle in the fourth quadrant = 360 - that = 318.5903779 degrees.
replace x in the equation with that to get 0 = 0, confirming that angle also satisfies the equation.
as i said before, i don't think you needed to know that, therefore your solution should be:
x = 180 degrees or x = 41.40962211 degrees.
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