Question

Tg(-225°)×cos120°=2/3sin300°-tg330°=cos^2 7/4п×ctg^2 2/3п=​

Answer 1

Ответ:

$1)\ \ tg(-225^\circ )\cdot cos120^\circ =-tg(270^\circ -45^\circ )\cdot cos(180^\circ -60^\circ )=\\\\=-tg45^\circ \cdot (-cos60^\circ )=\dfrac{1}{2}\\\\\\2)\ \dfrac{2}{3}\cdot sin300^\circ -tg330^\circ =\dfrac{2}{3}\cdot sin(360^\circ -60^\circ )-tg(360^\circ -30^\circ )=\\\\=\dfrac{2}{3}\cdot (-sin60^\circ )+tg30^\circ =-\dfrac{2}{3}\cdot \dfrac{\sqrt3}{2}-\dfrac{1}{\sqrt3}=-\dfrac{1}{\sqrt3}-\dfrac{1}{\sqrt3}=-\dfrac{2}{\sqrt3}=-\dfrac{2\sqrt3}{3}$

$3)\ \ cos^2\dfrac{7\pi }{4}\cdot ctg^2\dfrac{2\pi}{3}=cos^2\Big(2\pi -\dfrac{\pi}{4}\Big)\cdot ctg^2\Big(\pi -\dfrac{\pi}{3}\Big)=cos^2\dfrac{\pi}{4}\cdot ctg^2\dfrac{\pi}{3}=\\\\=\Big(\dfrac{\sqrt2}{2}\Big)^2\cdot \Big(\dfrac{1}{\sqrt3}\Big)^2=\dfrac{1}{2}\cdot \dfrac{1}{3}=\dfrac{1}{6}$

Answer 2

Let's break down the given trigonometric expressions step by step:

1. Tg(-225°): First, let's find the tangent of -225 degrees. Keep in mind that the tangent function has a period of 180 degrees, so we can find an equivalent angle within the range of -90 degrees to 90 degrees:

   -225° = -180° - 45°

   So, tg(-225°) = tg(-180° - 45°) = tg(-45°).

   The tangent of -45 degrees is -1, so tg(-225°) = -1.

2. cos120°: The cosine of 120 degrees can be found using the unit circle or reference angles. We know that cos(120°) = cos(180° - 60°). The cosine of 60 degrees is 0.5, and since the cosine function is negative in the second quadrant, cos(120°) = -0.5.

3. 2/3sin300°: The sine of 300 degrees can be found using reference angles. We know that sin(300°) = sin(360° - 60°). The sine of 60 degrees is √3/2, so sin(300°) = √3/2.

4. tg330°: The tangent of 330 degrees can be found using reference angles. We know that tg(330°) = tg(360° - 30°). The tangent of 30 degrees is 1/√3, so tg(330°) = 1/√3.

5. cos^2(7/4π): First, we need to convert 7/4π to degrees. Since π radians is equivalent to 180 degrees, we can calculate it as follows:

   7/4π = (7/4) * 180° = 315°

   Now, we can find cos^2(315°). The cosine of 315 degrees is -√2/2, so:

   cos^2(315°) = (-√2/2)^2 = 2/4 = 1/2

6. ctg^2(2/3π): First, convert 2/3π to degrees:

   2/3π = (2/3) * 180° = 120°

   Now, find the cotangent of 120 degrees. The cotangent is the reciprocal of the tangent, so ctg(120°) = 1/tg(120°). We already found tg(120°) to be 1/√3, so:

   ctg(120°) = 1/(1/√3) = √3

Now, let's put it all together:

• Tg(-225°) × cos120° = (-1) * (-0.5) = 0.5
• 2/3sin300° - tg330° = (2/3) * (√3/2) - (1/√3) = (√3/3) - (1/√3) = (√3 - 1) / √3
• cos^2(7/4π) × ctg^2(2/3π) = (1/2) * (√3) = √3/2

So, the final result is:

0.5 = (√3 - 1) / √3 ≠ √3/2