# 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 eparticularÃ„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

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2.

3.

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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 sin^{2} 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 a^{2}+b^{2}>=c^{2}.

## E. Homogeneous equations in *sin *and* cos*

Homogeneous equations are of the form:

unde

We divide through cos^{n}x and we get:

And if we note tg*x* = *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