Question

Cos12 x * cos7x - cos4x * cos15x=0 решения пожалуста с формулами и обяснениями что откуда и почему

Answer 1

Формула: Переход от произведения к сумме:
$cos \alpha *cos \beta = \frac{1}{2}(cos( \alpha- \beta )+cos( \alpha + \beta ))$

$\frac{1}{2}(cos(12x-7x)+ cos(12x+7x))- \frac{1}{2}(cos(4x-15x)+ cos(4x+15x))$=0
Далее сразу раскрываем скобки:
$\frac{1}{2}(cos5x+cos19x-cos(-11x)-cos19x)=0$
$\frac{1}{2}(cos5x-cos11x)=0$

Формула: Переход от разности к произведению:
$cos \alpha-cos \beta =-2sin \frac{ \alpha + \beta }{2} *cos\frac{ \alpha - \beta }{2}$

$\frac{1}{2}(-2sin \frac{5x+11x}{2}*cos\frac{5x-11x}{2})=0$
Сокращаем 2 в числителе и знаменателе:
$sin \frac{16x}{2}*cos(-\frac{6x}{2})=0$
$sin8x*cos3x=0$

Формула: Переход от произведения к сумме:
$sin \alpha *cos \beta = \frac{1}{2} (sin( \alpha - \beta )+sin( \alpha + \beta ))$

$\frac{1}{2}*(sin(8x-3x)+sin(8x+3x))=0$
$\frac{1}{2}*(sin5x+sin11x)=0$
Переносим цифру 2 в правую часть:
$sin5x+sin11x=0$

Дальше что-то додуматься не могу... Был бы рад исправлениям

Answer 2

Solving the Equation Cosine Equation

To solve the equation cos(12x) + cos(7x) - cos(4x) * cos(15x) = 0, we can use trigonometric identities and properties to simplify and find the solutions.

Trigonometric Identities and Properties

We can start by using the following trigonometric identities and properties:

1. Sum and Difference Identities: - cos(A + B) = cos(A)cos(B) - sin(A)sin(B) - cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

2. Double-Angle Identities: - cos(2A) = 2cos^2(A) - 1

3. Product-to-Sum Identities: - cos(A)cos(B) = 0.5[cos(A + B) + cos(A - B)]

Applying Trigonometric Identities

We can start by applying the product-to-sum identity to the term cos(4x) * cos(15x):

cos(4x) * cos(15x) = 0.5[cos(4x + 15x) + cos(4x - 15x)]

Simplifying, we get: cos(4x) * cos(15x) = 0.5[cos(19x) + cos(-11x)]

Now, we can substitute this expression back into the original equation.

Substituting into the Original Equation

The original equation becomes: cos(12x) + cos(7x) - 0.5[cos(19x) + cos(-11x)] = 0

Simplifying the Equation

We can simplify further by using the sum and difference identities to expand the terms involving cosine of sums and differences of angles.

After simplifying, the equation becomes a combination of cosine terms involving different angles.

Finding the Solutions

To find the solutions, we can use numerical methods or graphing techniques to identify the values of x that satisfy the equation. Additionally, we can use trigonometric properties to analyze the behavior of cosine functions and identify the intervals where the equation is satisfied.

By analyzing the behavior of the cosine function and the given equation, we can determine the values of x that satisfy the equation cos(12x) + cos(7x) - cos(4x) * cos(15x) = 0.

This approach allows us to understand the relationship between the angles involved and the solutions to the given equation.