Question

1 + sinx + cosx. Преобразовать в произведение Park01 ответил: 1 + sinx + cosx= (1+cos(x))+sin(x)=2cos^2(x/2) + 2*sin(x/2)*cos(x/2)= 2*cos(x/2)*(cos(x/2)+sin(x/2))=2*cos(x/2)*(cos(x/2)*sin(pi/4)+sin(x/2)*cos(pi/4))=2*cos(x/2)*sin(x/2+pi/4) Подскажите как вы получили 2*cos(x/2)*(cos(x/2)*sin(pi/4)+sin(x/2)*cos(pi/4))?

Answer 1

Для преобразования необходимо было вынести $\sqrt{2}$ за скобку. Так получилось бы выражение $\frac{1}{\sqrt{2}} = cos(\frac{\pi}{4}) = sin(\frac{\pi}{4})$, которое, очевидно, преобразуется в синус суммы или косинус разности.

Answer 2

Let's break down the steps to understand how the expression was simplified:

Starting expression: 1 + sin(x) + cos(x)

Step 1: Grouping We group the first two terms together and rewrite the expression as follows: 1 + sin(x) + cos(x) = (1 + cos(x)) + sin(x)

Step 2: Trigonometric identities Now, we use the trigonometric identity: sin(a + b) = sin(a)*cos(b) + cos(a)*sin(b)

In our expression, we can rewrite sin(x) as sin(x + π/4) using this identity: (1 + cos(x)) + sin(x) = (1 + cos(x)) + sin(x + π/4)

Step 3: Applying another trigonometric identity Next, we use the trigonometric identity: cos(2θ) = 2*cos^2(θ) - 1

We can see that 2θ = x + π/4, so θ = (x + π/4)/2. Now we substitute this in the identity above: cos(x + π/4) = 2*cos^2((x + π/4)/2) - 1

Step 4: Rearrange the terms Rearrange the identity to solve for 2cos^2((x + π/4)/2): 2cos^2((x + π/4)/2) = cos(x + π/4) + 1

Step 5: Substitute back into the expression Now, substitute back into our original expression: (1 + cos(x)) + sin(x + π/4) = 1 + cos(x + π/4) + sin(x + π/4) = 2cos^2((x + π/4)/2) + 2sin((x + π/4)/2)*cos((x + π/4)/2)

Step 6: Simplify Finally, simplify the expression: 2cos^2((x + π/4)/2) + 2sin((x + π/4)/2)cos((x + π/4)/2) = 2cos((x + π/4)/2) * (cos((x + π/4)/2) + sin((x + π/4)/2))

And that's how the expression 2cos(x/2)(cos(x/2)*sin(π/4) + sin(x/2)*cos(π/4)) was obtained from the original expression 1 + sin(x) + cos(x).