A trigonometric equation is any equation that includes a trigonometric
function. There are two basic types of trigonometric equations:
identities and conditional equations. Identities are equations that hold
for any angle. Conditional equations are equations that are solved only by
certain angles.
There are dozens of important trigonometric identities. Remember, the
identities below are true for any angle.
Eight Fundamental Identities
fundamental
csc(θ) = .


sec(θ) = .


cot(θ) = .


tan(θ) = .


cot(θ) = .


(sin(θ))^{2} + (cos(θ))^{2} = 1.


1 + (tan(θ))^{2} = (sec(θ))^{2}.


1 + (cot(θ))^{2} = (csc(θ))^{2}.


Cofunction Identities
cofunction
sin(  x) = cos(x).


cos(  x) = sin(x).


tan(  x) = cot(x).


cot(  x) = tan(x).


csc(  x) = sec(x).


sec(  x) = csc(x).


Negative Angle Identities
Sine, tangent, cosecant, and cotangent are
odd functions.
Cosine and secant are even
functions. These characteristics are evident in the negative angle identities.
negative
Double Angle Formulas
double
sin(2x) = 2 sin(x)cos(x).


cos(2x) = cos^{2}(x)  sin^{2}(x) = 1  2 sin^{2}(x) = 2 cos^{2}(x)  1.


tan(2x) = .


Half Angle Formulas
half
sin() = ±.


cos() = ±.


Addition Formulas
addition
sin(α + β) = sin(α)cos(β) + cos(α)sin(β).


cos(α + β) = cos(α)cos(β)  sin(α)sin(β).


tan(α + β) = .


Subtraction Formulas
subtraction
sin(α  β) = sin(α)cos(β)  cos(α)sin(β).


cos(α  β) = cos(α)cos(β) + sin(α)sin(β).


tan(α  β) = .


Product Formulas
product
sin(α)sin(β) =  (cos(α + β)  cos(α  β)).


cos(α)cos(β) = (cos(α + β) + cos(α  β)).


sin(α)cos(β) = (sin(α + β) + sin(α  β)).


cos(α)sin(β) = (sin(α + β)  sin(α  β)).


Sum and Difference Formulas
sumdifference
sin(α) + sin(β) = 2 sin(cos(.


cos(α) + cos(β) = 2 cos(cos(.


sin(α)  sin(β) = 2 cos(sin(.


cos(α)  cos(β) =  2 sin(sin(.


There is no single method for solving trigonometric equations. A few techniques
come in handy, though. 1) Resolve everything into terms of sine and cosine,
then cancel everything possible. 2) Manipulate the equation with factoring and
other algebraic techniques to create trigonometric identities that can be
simplified. 3) If a solution can't be reached, try graphing the equation to
solve it.
In every trigonometric equation, there will be either no solutions or an
infinite number of solutions. The reason for this is that the trigonometric
functions are periodic. It is customary to
only list the solutions
x
where
0≤x < 2Π
or, if the period involved
is different from
2Π
, to describe all solutions.
Solving triangles is one of the key applications of trigonometric functions. To
see a discussion of solving triangles using trigonometry, see Solving Right
Triangles and Solving Oblique
Triangles.