Example 1: Solve the trigonometric equation.(find all solutions)
2 cos x + 2 = 3
Solution to example 1
solve for cos(x)
cos x = 1/2
solve for x by finding all values in the
interval [0 , 2pi) that satisfy the above trigonometric equation.
In this case, with cosine positive and equal to 1 / 2, there
are two values: one in the first quadrant of the unit circle.
x1 = pi / 3
and a second one in the fourth quadrant (see the two solutions in unit circle in figure below).
x2 = 2*pi - pi / 3 = 5*pi / 3
find all solutions using the fact that of cos x has a period of 2pi
x1 = pi / 3 + 2*k*pi
x2 = 5*pi / 3 + 2*k*pi
where k is any integer
conclusion: There is an infinite number of solutions which
can be generated by giving different values to k.
Example 2: Find the solutions in the interval [0 , 2pi) for the trigonometric equation
-5 cos 2x + 9 sin x = -3
Solution to example 2
change cos 2x to 1 - sin 2x
-5(1 - sin 2x) + 9 sin x = -3
multiply factors and group to obtain
5 sin2x + 9 sin x -2 = 0
let u = sinx and substitute to obtain an
5 u2 + 9 u - 2 = 0
use any method to solve for u.
By the quadratic formula, we obtain two solutions u1 and u2
u1 = [ -9 - sqrt(121) ] / 10 = 1 / 5 = -2
u2 = [ -9 + sqrt(121) ] / 10 = 0.2
we now solve the equation for x
u1 = sin x = -2
the above equation has no solutions for x since -2 is not in the range of values of sin(x);
-1 <= sin x <= 1.
u2 = sin x = 0.2
sin x is positive in the first and second
quadrants of the unit circle. There are two
solutions to the above equation in [0 , 2 pi)(see unit circle below)
solution in the first quadrant
x1 = arcsin 0.2
arcsin 0.2 is the inverse sine function
and it can be approximated using a calculator)
solution in the second quadrant
x2 = pi - arcsin 0.2
conclusion: There are two solutions to the given equation.
Example 3: Find all solutions for the trigonometric equation
cot x cos 2x = cot x
Solution to example 3
Note that many of the techniques used in solving
algebraic equations are also used to solve trigonometric
subtract cot x from both sides of the equation and simplify
cot x cos 2x - cot x = 0
Factor cot x
cot x (cos 2x - 1) = 0
Setting each factor in the above trigonometric equation to zero, we obtain two equations.
cot x = 0 and cos 2x - 1 = 0
The solutions to equation cot x = 0 are given by
x = pi / 2 + k*pi , k is am integer.
- Equation cos 2x - 1 = 0 gives
cos x = 1 and cos x = -1
The solutions to the above equations are given by
x = 2k*pi and x = (2k + 1)*pi where k is an integer
HOWEVER the above cannot be solution to the given equation since cot x is undefined for x = 2k*pi and x = (2k + 1)*pi.
conclusion: The solutions to the given equation are.
x = pi / 2 + k*pi where k is an integer.