Precalculus: Trigonometric Functions
Tigonometric Equations
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) =  .
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sec(θ) =  .
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cot(θ) =  .
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tan(θ) =  .
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cot(θ) =  .
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| (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).
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cos( - x) = sin(x).
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tan( - x) = cot(x).
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cot( - x) = tan(x).
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csc( - x) = sec(x).
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sec( - x) = csc(x).
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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
| sin(- θ) = - sin(θ). |
| cos(- θ) = cos(θ). |
| tan(- θ) = - tan(θ). |
| csc(- θ) = - csc(θ). |
| sec(- θ) = sec(θ). |
| cot(- θ) = - cot(θ). |
Double Angle Formulas
double
| sin(2x) = 2 sin(x)cos(x). |
| cos(2x) = cos2(x) - sin2(x) = 1 - 2 sin2(x) = 2 cos2(x) - 1. |
tan(2x) =  .
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Half Angle Formulas
half
sin( ) = ± .
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cos() = ± .
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tan() = ± =  =  .
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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) =  .
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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) =  .
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Product Formulas
product
sin(α)sin(β) = -  (cos(α + β) - cos(α - β)).
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cos(α)cos(β) = (cos(α + β) + cos(α - β)).
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sin(α)cos(β) = (sin(α + β) + sin(α - β)).
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cos(α)sin(β) = (sin(α + β) - sin(α - β)).
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Sum and Difference Formulas
sumdifference
sin(α) + sin(β) = 2 sin( cos( .
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cos(α) + cos(β) = 2 cos(cos(.
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sin(α) - sin(β) = 2 cos(sin(.
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cos(α) - cos(β) = - 2 sin(sin(.
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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.
