Question

Найти минимальное и максимальное значения cos a - sin a

Answer 1

Ответ:

$-\sqrt{2}$ и $\sqrt{2}$ соответственно.

Пошаговое объяснение:

$cos(a) - sin(a) = \sqrt{2}(\frac{1}{\sqrt{2}}cos(a) - \frac{1}{\sqrt{2}}sin(a)) = \sqrt{2} cos(a + \frac{\pi}{4})\\cos(a + \frac{\pi}{4}) \in [-1; 1] \Rightarrow \sqrt{2} cos(a + \frac{\pi}{4}) \in [-\sqrt{2}; \sqrt{2}]$

Answer 2

To find the minimum and maximum values of the expression cos(a) - sin(a), we can analyze the behavior of the trigonometric functions involved.

The expression cos(a) - sin(a) can be rewritten as a single trigonometric function using angle sum/difference identities. Let's use the identity: cos(π/4 - a) = cos(π/4) * cos(a) + sin(π/4) * sin(a) = (sqrt(2)/2) * cos(a) + (sqrt(2)/2) * sin(a)

Now, the expression cos(a) - sin(a) becomes: (sqrt(2)/2) * cos(a) + (sqrt(2)/2) * sin(a) - sin(a)

Let's further simplify this expression: (sqrt(2)/2) * cos(a) - (sqrt(2)/2) * sin(a)

Now, we can see that the expression is in the form: A * cos(a) + B * sin(a), where A = sqrt(2)/2 and B = -sqrt(2)/2.

The maximum value of A * cos(a) + B * sin(a) is sqrt(A^2 + B^2), and the minimum value is -sqrt(A^2 + B^2).

So, in our case, the maximum value of cos(a) - sin(a) is sqrt((sqrt(2)/2)^2 + (-sqrt(2)/2)^2) = sqrt(1/2) = sqrt(2)/2 ≈ 0.707, and the minimum value is -sqrt(1/2) = -sqrt(2)/2 ≈ -0.707.

Therefore, the minimum value of cos(a) - sin(a) is approximately -0.707, and the maximum value is approximately 0.707.