Question

Вычислить cos(-45 °)+sin(315 °)=tan(-855 °)

Answer 1

Cos45+sin(360-45)=-tg(855)
корень из 2\2+корень из 2\2=-tg(5*180-45)
корень из 2\2+корень из 2\2=tg1

Answer 2

To solve this, we'll first find the values of cos(-45°) and sin(315°), and then verify if they satisfy the equation cos(-45°) + sin(315°) = tan(-855°). Keep in mind that trigonometric functions are periodic, so we can adjust the angles to their equivalent values within the first revolution (0° to 360°).

1. cos(-45°): The cosine function is an even function, meaning cos(-x) = cos(x). So, cos(-45°) = cos(45°).

2. sin(315°): The sine function is an odd function, meaning sin(-x) = -sin(x). So, sin(315°) = -sin(-45°) = -(-sin(45°)) = sin(45°).

Now, let's find the values of cos(45°) and sin(45°):

To do this, let's consider a right-angled triangle with angles 45°, 45°, and 90°, where the two sides adjacent to the 45° angles are equal in length (let's call them 'a'), and the hypotenuse is the square root of 2 times the length of 'a'.

3. cos(45°) = adjacent side / hypotenuse = 'a' / (√2 * 'a') = 1 / √2 = √2 / 2

4. sin(45°) = opposite side / hypotenuse = 'a' / (√2 * 'a') = 1 / √2 = √2 / 2

Now, let's verify if cos(-45°) + sin(315°) equals tan(-855°):

cos(-45°) + sin(315°) = cos(45°) + sin(45°) = √2 / 2 + √2 / 2 = √2

tan(-855°):

We know that tan(x) is periodic with a period of 180°. So, tan(-855°) is the same as tan(-855° + 6 * 180°) = tan(135°).

Now, in the unit circle, at 135°, the values of cos(135°) and sin(135°) are:

cos(135°) = -√2 / 2 sin(135°) = √2 / 2

So, tan(135°) = sin(135°) / cos(135°) = (√2 / 2) / (-√2 / 2) = -1

Since √2 ≠ -1, the equation cos(-45°) + sin(315°) = tan(-855°) is not true.