How to solve trigonometric equations

How to solve trigonometric equations?

It's no secret that success or failure in the processsolution of almost any problem, mainly depends on the correctness of determining the type of the given equation, and also on the correctness of the reproduction of the sequence of all stages of its solution. However, in the case of trigonometric equations, it is not at all difficult to determine the fact that the equation is trigonometric. But in the process of determining the sequence of actions that should lead us to the right answer, we can face certain difficulties. Let's figure out how to solve trigonometric equations correctly from the very beginning.

The solution of trigonometric equations

In order to solve the trigonometric equation, you need to try to fulfill the following points:

  • We give all the functions that enter our equation to "equal angles";
  • It is necessary to bring the given equation to "identical functions";
  • We decompose the left side of the given equation into factors or other necessary components.


Method 1. It is necessary to solve such equations in two stages. First, we transform the equation in order to get its simplest (simplified) form. The equation: Cosx = a, Sinx = a and the like, are called the simplest trigonometric equations. The second step is to solve the simplest equation obtained. It should be noted that the simplest equation can be solved by an algebraic method, which is well known to us from the school course of algebra. It is also called the substitution and variable substitution method. Using the reduction formulas, you first need to convert, then make a replacement and then find the roots.

Next, we need to expand our equation into possiblemultipliers, for this it is necessary to transfer all members to the left and then can be factorized. Now we need to bring this equation to a homogeneous one, in which all terms are equal to one degree, and the cosine and sine have the same angle.

Before solving trigonometricequation, we need to transfer its terms to the left side, taking them from the right, and then we take all the common denominators in brackets. Equate our parentheses and multipliers to zero. Our equated brackets are a homogeneous equation with a reduced degree, which must be divided into sin (cos) to the highest degree. Now solve the algebraic equation that was obtained, in relation to tan.

Method 2. Another method by which one can solve the trigonometric equation is to go to the half-angle. For example, solve the equation: 3sinx-5cosx = 7.

We need to go to the half-corner, in ourThe case is: 6sin (x / 2) * cos (x / 2) - 5cos² (x / 2) + 5sin² (x / 2) = 7sin² (x / 2) + 7cos² (x / 2). And then, all members in one part (for convenience it is better to choose the right one) and proceed to solving the equation.

If necessary, you can enter an auxiliary angle. This is done in the case where it is necessary to replace the integer value sin (a) or cos (a) and the sign "a" just acts as an auxiliary angle.

The product of the sum

How to solve trigonometric equations,using the product in the amount? A method known as the transformation of a product into a sum can also be used to solve such equations. In this case, it is necessary to use the formulas corresponding to the equation.

For example, we have the equation: 2sinx * sin3x = cos4x

We need to solve this problem by converting the left-hand side into a sum, namely:

cos 4x-cos8x = cos4x,

cos8x = 0,

8x = p / 2 + pk,

x = p / 16 + pk / 8.


If the above methods are not suitable, and you are allyet do not know how to solve the simplest trigonometric equations, you can use another method - a universal substitution. Using it, you can convert an expression and make a replacement. For example: Cos (x / 2) = u. Now we can solve the equation with the available parameter u. And having received the desired result, do not forget to translate this value into the opposite.

Many "experienced" students are advised to apply forsolving equations to people in online mode. How to solve the trigonometric equation online, you ask. For the online solution of the problem, you can refer to the relevant forums where you can be helped by advice or in solving the problem. But best of all, still try to do your own.

Skills and skills in solving trigonometricequations are very important and useful. Their development will require considerable effort from you. Many problems of physics, stereometry, etc. are connected with the solution of such equations. And the process of solving such problems involves the availability of skills and knowledge that can be acquired while studying the elements of trigonometry.

Teach trigonometric formulas

In the process of solving the equation, you canencounter the need to use any formula from trigonometry. You can, of course, start looking for it in your textbooks and cribs. And if these formulas are put off in your head, you will not only save your nerves, but also greatly facilitate your task without wasting time searching for the necessary information. Thus, you will have the opportunity to think through the most rational way to solve the task.

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