If cot3x = 4/3 (where 0 <x<30 ), then the value of 3cosecx - 4secx is
Answers
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.
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:
2
sin
(
4
x
−
π
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
(
4
x
−
π
3
)
by the single letter
S
. Now we need to find
S
to make
2
S
=
1
. Simple! Make
S
=
1
2
So a solution will need to make
sin
(
4
x
−
π
3
)
=
1
2
Step 2: The 'angle' in this equation is
(
4
x
−
π
3
)
. For the moment, let's call that
θ
. We need
sin
θ
=
1
2
There are infinitely many such