Question

1-8sin^2 π/16 × cos^2 π/16

Answer 1

Ответ:

$\frac{\sqrt{2}}{2}$

Пошаговое объяснение:

$1-8sin^2\frac{\pi}{16}cos^2\frac{\pi}{16}=1-2(2sin\frac{\pi}{16}cos\frac{\pi}{16})^2=\\\\=1-2*(sin\frac{\pi}{8})^2= sin^2\frac{\pi}{8}+cos^2\frac{\pi}{8}-2sin^2\frac{\pi}{8}=\\\\=cos^2\frac{\pi}{8}-sin^2\frac{\pi}{8}=cos(2*\frac{\pi}{8})=cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$

*** Для решения использованы формулы:

$2*sina*cosa=sin2a\\1=cos^2a+sin^2a\\cos^2a-sin^2a=cos2a$

Answer 2

To calculate the expression $1 - 8\sin^2(\frac{\pi}{16}) \times \cos^2(\frac{\pi}{16})$, we'll start by evaluating $\sin^2(\frac{\pi}{16})$ and $\cos^2(\frac{\pi}{16})$ and then substitute them into the given expression.

Let's calculate the values step by step:

1. $\sin(\frac{\pi}{16})$ and $\cos(\frac{\pi}{16})$:

We can use the half-angle formulas to find these values:

$$\sin^2(\frac{\pi}{16}) = \frac{1 - \cos(\frac{\pi}{8})}{2}$$

$$\cos^2(\frac{\pi}{16}) = \frac{1 + \cos(\frac{\pi}{8})}{2}$$

2. Substitute the values:

$$1 - 8\sin^2(\frac{\pi}{16}) \times \cos^2(\frac{\pi}{16}) = 1 - 8\left(\frac{1 - \cos(\frac{\pi}{8})}{2}\right) \times \left(\frac{1 + \cos(\frac{\pi}{8})}{2}\right)$$

3. Substitute the values of $\cos(\frac{\pi}{8})$:

$\cos(\frac{\pi}{8}) \approx 0.9239$ (approximately)

4. Plug in the values and calculate:

$$\begin{align*} 1 - 8\left(\frac{1 - 0.9239}{2}\right) \times \left(\frac{1 + 0.9239}{2}\right) &\approx 1 - 8 \times 0.03805 \times 0.96195 \\ &\approx 1 - 0.2933 \\ &\approx 0.7067 \, \text{(rounded to four decimal places)} \end{align*}$$