Question

Упростить √2cosх-2 cos(pi/4-x)/ 2sin(pi/6+x)-√3 sinx

Answer 1

$\frac{\sqrt{2}cosx-2cos(\frac{\pi}{4}-x)}{2sin(\frac{\pi}{6}+x)-\sqrt{3}sinx}=\frac{\sqrt{2}cosx-2cos\frac{\pi}{4}cosx+2sin\frac{\pi}{4}sinx}{2sin\frac{\pi}{6}cosx+2sinxcos\frac{\pi}{6}-\sqrt{3}sinx}=\frac{\sqrt{2}cosx-\sqrt{2}cosx+\sqrt{2}sinx}{cosx+\sqrt{3}sinx-\sqrt{3}sinx}=\\\\=\frac{\sqrt{2}sinx}{cosx}=\sqrt{2}tgx$

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$cos(\alpha-\beta)=cos\alpha \cdot cos\beta+sin\alpha \cdot sin\beta\\ sin(\alpha+\beta)=sin\alpha \cdot cos\beta+sin\beta \cdot cos\alpha \\ \\ sin\frac{\pi}{4}=cos\frac{\pi}{4}=\frac{\sqrt{2}}{2} \\ 2sin\frac{\pi}{4}=2cos\frac{\pi}{4}=\sqrt{2} \\ \\ sin\frac{\pi}{6}=\frac{1}{2} \\ 2sin\frac{\pi}{6}=1 \\ \\ cos\frac{\pi}{6}=\frac{\sqrt{3}}{2} \\ 2cos\frac{\pi}{6}=\sqrt{3}$

Answer 2

на фото.....................

Answer 3

To simplify the expression, we'll start by working on each term step by step:

1. Simplify √2cos(x):

√2cos(x) = √2 * cos(x)

2. Simplify -2cos(π/4 - x):

Using the trigonometric identity: cos(π/4 - x) = cos(π/4) * cos(x) + sin(π/4) * sin(x)

cos(π/4) = 1/√2 and sin(π/4) = 1/√2

So, cos(π/4 - x) = (1/√2) * cos(x) + (1/√2) * sin(x) = (cos(x) + sin(x)) / √2

Therefore, -2cos(π/4 - x) = -2 * (cos(x) + sin(x)) / √2 = -√2 * (cos(x) + sin(x))

3. Simplify 2sin(π/6 + x):

Using the trigonometric identity: sin(π/6 + x) = sin(π/6) * cos(x) + cos(π/6) * sin(x)

sin(π/6) = 1/2 and cos(π/6) = √3/2

So, sin(π/6 + x) = (1/2) * cos(x) + (√3/2) * sin(x)

Therefore, 2sin(π/6 + x) = 2 * ((1/2) * cos(x) + (√3/2) * sin(x)) = cos(x) + √3 * sin(x)

Now the expression becomes:

√2cos(x) - 2cos(π/4 - x) / 2sin(π/6 + x) - √3sin(x)

Substitute the simplified terms:

√2cos(x) - (-√2 * (cos(x) + sin(x))) / (cos(x) + √3 * sin(x))

Now, to simplify further, let's combine the fractions by finding the common denominator, which is (cos(x) + √3 * sin(x)):

√2cos(x) + √2(cos(x) + sin(x)) / (cos(x) + √3 * sin(x))

Now, combine the terms in the numerator:

√2cos(x) + √2cos(x) + √2sin(x) / (cos(x) + √3 * sin(x))

Combine like terms:

2√2cos(x) + √2sin(x) / (cos(x) + √3 * sin(x))

And that's the simplified expression.