Question

Cos120°-ctg 60°+sin 90°cos 135°

Answer 1

Ответ:

$cos120^\circ -ctg60^\circ +sin90^\circ \cdot cos135^\circ =cos(180^\circ -60^\circ )-\dfrac{\sqrt3}{3}+1\cdot cos(90^\circ +45^\circ )=\\\\=-cos60^\circ -\dfrac{\sqrt3}{3}-sin45^\circ =-\dfrac{1}{2}-\dfrac{\sqrt3}{3}-\dfrac{\sqrt2}{2}=-\dfrac{3+2\sqrt3+3\sqrt2}{6}$

Answer 2

Let's break down each part of the expression and calculate it step by step:

1. Cosine of 120 degrees (cos 120°): cos 120° = -1/2

2. Cotangent of 60 degrees (ctg 60°) is the reciprocal of the tangent: cot 60° = 1/tan 60°

   To find tan 60°, we can use the fact that tan 60° = sqrt(3): cot 60° = 1/sqrt(3)

3. Sine of 90 degrees (sin 90°): sin 90° = 1

4. Cosine of 135 degrees (cos 135°): cos 135° = -sqrt(2)/2

Now, let's substitute these values back into the original expression and calculate it:

-1/2 - (1/sqrt(3)) + 1 * (-sqrt(2)/2)

Now, let's simplify further:

-1/2 - (1/sqrt(3)) - sqrt(2)/2

To add these fractions, we need a common denominator, which is sqrt(3):

(-sqrt(3)/2) - (1/sqrt(3)) - (sqrt(2)/2)

Now, combine the fractions:

(-sqrt(3) - 1 - sqrt(2))/sqrt(3)

So, the simplified expression is:

(-sqrt(3) - 1 - sqrt(2))/sqrt(3)