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^{ 2}x + 9 sin x = 3
Solution to example 2

change cos^{ 2}x to 1  sin^{ 2}x
5(1  sin^{ 2}x) + 9 sin x = 3

multiply factors and group to obtain
5 sin^{2}x + 9 sin x 2 = 0

let u = sinx and substitute to obtain an
quadratic equation.
5 u^{2} + 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
 and
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^{ 2}x = cot x
Solution to example 3

subtract cot x from both sides of the equation and simplify
cot x cos^{ 2}x  cot x = 0

Factor cot x
cot x (cos^{ 2}x  1) = 0

Setting each factor in the above trigonometric equation to zero, we obtain two equations.
cot x = 0 and cos^{ 2}x  1 = 0

The solutions to equation cot x = 0 are given by
x = pi / 2 + k*pi , k is am integer.
 Equation cos^{ 2}x  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.
Note that many of the techniques used in solving
algebraic equations are also used to solve trigonometric
equations.