Question

1)ctg π/3-π/42)2sin(π/4+α)=cos^2α+sin^2α​

Answer 1

Ответ:

1) корень из 3    /3 -5/4

2)остаётся решение и дальше с этим ничего не сделать

2 корня из 2 аcos^ умножить cos (a^) +2корня из 2 acos^ умножить sin (a)

+2an^ si

Пошаговое объяснение:

Answer 2

Let's solve each of these trigonometric equations step by step:

1. ctg(π/3 - π/4):

First, let's simplify the expression by finding the common denominator for the angles inside the cotangent function:

π/3 = (4π)/(12) π/4 = (3π)/(12)

Now, we can rewrite the expression as:

ctg((4π)/(12) - (3π)/(12))

Simplifying the angles:

ctg(π/(12))

Recall that cotangent is the reciprocal of the tangent function. To find the cotangent of an angle, we can use the identity:

ctg(x) = 1 / tan(x)

Using this identity, we can rewrite the equation as:

1 / tan(π/(12))

Now, let's calculate the tangent of π/12:

tan(π/12)

To do this, we can use the tangent half-angle formula:

tan(x/2) = (1 - cos(x)) / sin(x)

Plugging in x = π/6:

tan(π/12) = (1 - cos(π/6)) / sin(π/6)

cos(π/6) = √3/2 sin(π/6) = 1/2

tan(π/12) = (1 - √3/2) / (1/2) = 2 - √3

Now, let's substitute this back into the original equation:

1 / (2 - √3)

To rationalize the denominator, we can multiply both the numerator and denominator by the conjugate of the denominator:

1 / (2 - √3) * (2 + √3) / (2 + √3)

Simplifying:

= (2 + √3) / (4 - 3) = (2 + √3)

So, ctg(π/3 - π/4) = 2 + √3.

2. 2sin(π/4 + α) = cos^2(α) + sin^2(α)

Let's simplify this equation step by step:

First, expand the square of the cosine and sine:

2sin(π/4 + α) = cos^2(α) + sin^2(α) 2sin(π/4 + α) = (1 - sin^2(α)) + sin^2(α)

Now, simplify further:

2sin(π/4 + α) = 1

Divide both sides by 2:

sin(π/4 + α) = 1/2

We know that the sine function takes a value of 1/2 at π/6, so we can write:

π/4 + α = π/6 + 2πn

where n is an integer representing the number of complete cycles. Now, solve for α:

α = π/6 - π/4 + 2πn

Simplifying:

α = (π - 3π/2 + 12πn)/12 α = (π - 3π/2 + 12πn)/12

So, the solution for α is α = (π - 3π/2 + 12πn)/12, where n is an integer.