Solving Trigonometric Equations
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Key Questions

Examples of trig expressions:
# f(x) = sin 2x + cos x; #
#f(x) = sin x + sin 2x + sin 3x#
Examples of trig equations:# f(x) = sin 2x + cos x = 0#
# f(x) = sin x + sin 2x + sin 3x = 0#
Examples of trig inequalities#f(x) = sin 2x + cos x > 0#
# f(x) = sin x + sin 2x + sin 3x < 0# Use trig Transformation Identities to transform these above trig expressions into trig basic expressions, or expressions in simplest form.
Example: Transform#f(x) = sin 2x + cos x# . Use Identity#(sin 2a = 2 sin a*cos a)# to transform#f(x).#
#f(x) = 2*sin x*cos x + cos x = cos x*(2sin x + 1)#
This is f(x) expressed in simplest form.
Trig equation in simplest form:#f(x) = cos x*(sin 2x + 1) = 0#
Trig inequality in simplest form:#f(x) = cos x*(2sinx + 1) > 0# 
Solving concept. To solve a trig equation, transform it into one, or many, basic trig equations. Solving a trig equation, finally, results in solving various basic trig equations.
There are 4 main basic trig equations:
sin x = a; cos x = a; tan x = a; cot x = a.
Exp. Solve sin 2x  2sin x = 0
Solution. Transform the equation into 2 basic trig equations:
2sin x.cos x  2sin x = 0
2sin x(cos x  1) = 0.
Next, solve the 2 basic equations: sin x = 0, and cos x = 1.
Transformation process.
There are 2 main approaches to solve a trig function F(x).
1. Transform F(x) into a product of many basic trig functions.
Exp. Solve F(x) = cos x + cos 2x + cos 3x = 0.
Solution. Use trig identity to transform (cos x + cos 3x):
F(x) = 2cos 2x.cos x + cos 2x = cos 2x(2cos x + 1 ) = 0.
Next, solve the 2 basic trig equations.
2. Transform a trig equation F(x) that has many trig functions as variable, into a equation that has only one variable. The common variables to be chosen are: cos x, sin x, tan x, and tan (x/2)
Exp Solve#sin ^2 x + sin^4 x = cos^2 x#
Solution. Call cos x = t, we get
#(1  t^2)(1 + 1  t^2) = t^2# .
Next, solve this equation for t.
Note . There are complicated trig equations that require special transformations. 
As a general description, there are 3 steps. These steps may be very challenging, or even impossible, depending on the equation.
Step 1: Find the trigonometric values need to be to solve the equation.
Step 2: Find all 'angles' that give us these values from step 1.
Step 3: Find the values of the unknown that will result in angles that we got in step 2.(Long) Example
Solve:#2sin(4x pi/3)=1# Step 1: The only trig function in this equation is
#sin# .
Sometimes it is helpful to make things look simpler by replacing, like this:
Replace#sin(4x pi/3)# by the single letter#S# . Now we need to find#S# to make#2S=1# . Simple! Make#S=1/2#
So a solution will need to make#sin(4x pi/3)=1/2# Step 2: The 'angle' in this equation is
#(4x pi/3)# . For the moment, let's call that#theta# . We need#sin theta = 1/2#
There are infinitely many such#theta# , we need to find them all.Every
#theta# that makes#sin theta = 1/2# is coterminal with either#pi/6# or with#(5 pi)/6# . (Go through one period of the graph, or once around the unit circle.)
So#theta# Which, remember is our short way of writing#4x pi/3# must be of the form:#theta = pi/6+2 pi k# for some integer#k# or of the form#theta = (5 pi)/6 +2 pi k# for some integer#k# .Step 3:
Replacing#theta# in the last bit of step 2, we see that we need one of:#4x pi/3 = pi/6+2 pi k# for integer#k#
or#4x pi/3 = (5 pi)/6+2 pi k# for integer#k# .Adding
# pi/3# in the form#(2 pi)/6# to both sides of these equations gives us:
#4x = (3 pi)/6+2 pi k = pi/2+2 pi k# for integer#k# or
#4x = (7 pi)/6+2 pi k# for integer#k# .Dividing by
#4# (multiplying by#1/4# ) gets us to:#x= pi/8+(2pi k)/4# or
#x=(7 pi)/24+(2 pi k)/4# for integer#k# .We can write this in simpler form:
#x= pi/8+pi/2 k# or
#x=(7 pi)/24+pi/2 k# for integer#k# .Final note The Integer
#k# could be a positive or negative whole number or 0. If#k# is negative, we're actually subtracting from the basic solution. 
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