Trigonometry

Trigonometric Equations

Trigonometric equations are those equations the unknown is the composition of the arguments of trigonometric functions.

Trigonometric equations are equations transcendent. That is, it causes a eparticularĂ„Æ’ solutions and then write the general solution is expressed in terms of a parameter and of the period of the trigonometric function.

 

A.     Elementary trigonometric equations

1.      

2.      

3.      

4.      

 

B.     Equations of the form f(g(x)) = f(h(x)):

1.      

2.      

3.      

4.      

 

C.     Trigonometric equations that are solved using algebraic equations

1.      

2.      

3.      

4.      

 

Equations of this type are reduced to solving the equation and solve a trigonometric equation or two after one of the basic notations: sin x = t, cos  x = t, tg x = t, ctg x = t.

5.       Using the formula cos 2x = 1- 2 sin2 x we can obrain an first degree equation.

6.       Because  since  , we can obtain the equation  , i.e. an equation of type 3.

 

D.    Linear trigonometric equations in sin and cos with the form: a sin x + b cos x + c = 0, with a and b not zero.

 

The Method 1.

Check if the equation has solutions of the form: .

It notes .

We have the formulas

And thus, we obtain the equation , with the condition .

The Method 2.

We write the substitutions , and we solve the equations system

The Method 3.

We make the substitution and we get

The equation has solutions if a2+b2>=c2.

 

E.     Homogeneous equations in sin and cos

Homogeneous equations are of the form:

    unde

 

We divide through cosnx and we get:

And if we note tgx = y then we get:

F.      Equations symmetrical with sin and cos:

1.      

We have 3 ways of solving:

Method 1. I note   and I use the formulas , .

Method 2. I note  and I get the equation which it resolves with condition  .

Method 3. I note and the equation becomes

 

G.    Other types of equations

 

1.      

We rise the identity at the powers n, n-1, …., 3,2

Then we note sin 2x = t and we get an equation rank  n with the unknown  t.

 

Th  The equations having the form as

.

We use the formulas

,

3.       Equations containing products form:

Convert products in the amounts or differences sinus (sine) or cosines (cosine).

Keywords: trigonometry, trigonometric equations, mathematical formulas, sine, cosine, tangent

 

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