How to solve trigonometric equations
Video: Solving Trigonometric Equations Using Identities, Multiple Angles, By Factoring, General Solution 2024, July
Trigonometric equations are equations that contain the trigonometric functions of an unknown argument (for example: 5sinx-3cosx = 7). To learn how to solve them, you need to know some methods for this.
Instruction manual
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The solution to such equations consists of two stages.
The first is the transformation of the equation to obtain its simplest form. The simplest trigonometric equations are as follows: Sinx = a; Cosx = a etc.
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The second is the solution to the simplest trigonometric equation obtained. There are basic methods for solving equations of this kind:
Solution by the algebraic method. This method is well known from school, with a course in algebra. In another name, the method of variable replacement and substitution. Using the reduction formulas, we transform, make a replacement, and then find the roots.
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Factorization of the equation. First, transfer all terms to the left and factor them.
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Bringing the equation to a homogeneous one. Homogeneous equations are called equations if all members of the same degree and sine, cosine of the same angle.
To solve it, you should: first transfer all its members from the right side to the left side; put all common factors out of brackets; equate factors and brackets to zero; equal brackets give a homogeneous equation of a lesser degree, which should be divided into cos (or sin) in a higher degree; solve the resulting algebraic equation for tan.
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The next method is the transition to the half corner. For example, solve the equation: 3 sin x - 5 cos x = 7.
Go to the half angle: 6 sin (x / 2) · cos (x / 2) - 5 cos ² (x / 2) + 5 sin ² (x / 2) = 7 sin ² (x / 2) + 7 cos ² (x / 2), after which we reduce all terms to one part (preferably to the right) and solve the equation.
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The introduction of the auxiliary angle. When we replace the integer value cos (a) or sin (a). Sign "a" is an auxiliary angle.
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The method of converting a work into a sum Here you must use the appropriate formulas. For example, given: 2 sin x sin 3x = cos 4x.
We solve it by converting the left side to a sum, that is:
cos 4x - cos 8x = cos 4x, cos 8x = 0, 8x = p / 2 + pk, x = p / 16 + pk / 8.
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The latter method, called universal substitution. We transform the expression and make a replacement, for example, Cos (x / 2) = u, after which we solve the equation with the parameter u. Upon receipt of the result, we translate the value to the opposite.