Question

1)arccos(-1)-arccos 1/2 -3 arccos(- √3/2) 2)2сos t=1 3)-2cos t=0 4)cos (arcctg √3) 5)arcctg (ctg 2П/3) 6)ctg x=-0,5 7)ctg x=0

Answer 1

============== 1 ==============

$arccos(-1)-arccos \frac{1}{2} -3 arccos(- \frac{ \sqrt{3} }{2} )= \\ = \pi -arccos1-arccos \frac{1}{2}-3( \pi -arccos \frac{ \sqrt{3} }{2}) = \pi -0- \frac{ \pi }{3} -3( \pi - \frac{ \pi }{6})=\\ = \frac{2 \pi }{3}- \frac{5 \pi }{2} =\frac{4 \pi }{6}- \frac{15 \pi }{6} =- \frac{ 11\pi }{6}$

============== 2 ==============

$> ============== 3 ============== <img src=$

============== 4 ==============

$cos (arcctg \sqrt{3} )=cos \frac{ \pi }{6} = \frac{ \sqrt{3} }{2}$

============== 5 ==============

$arcctg (ctg \frac{2 \pi }{3} )= \frac{2 \pi }{3}$

============== 6 ==============

$> ============== 7 ============== <img src=$ 0 0

Answer 2

Let's solve each of the given trigonometric equations and expressions step by step:

1. arccos(-1) - arccos(1/2) - 3arccos(-√3/2): arccos(-1) = π arccos(1/2) = π/3 arccos(-√3/2) = 2π/3

Substituting these values: π - π/3 - 3(2π/3) = π - π/3 - 2π = -π - π/3 = -4π/3

Therefore, arccos(-1) - arccos(1/2) - 3arccos(-√3/2) simplifies to -4π/3.

2. 2cos(t) = 1: Dividing both sides by 2: cos(t) = 1/2

The solutions for this equation occur at angles where the cosine value is 1/2. The standard angles where the cosine is positive 1/2 are π/3 and 5π/3. Adding the period of cosine, we can generalize the solutions as: t = π/3 + 2πn, where n is an integer.

3. -2cos(t) = 0: Dividing both sides by -2: cos(t) = 0

The solutions for this equation occur at angles where the cosine value is 0. The standard angles where the cosine is 0 are π/2 and 3π/2. Adding the period of cosine, we can generalize the solutions as: t = π/2 + πn, where n is an integer.

4. cos(arcctg(√3)): arcctg(√3) = π/6 (inverse cotangent of √3)

cos(π/6) = √3/2

Therefore, cos(arcctg(√3)) simplifies to √3/2.

5. arcctg(ctg(2π/3)): ctg(2π/3) = -1/√3 (cotangent of 2π/3)

arcctg(-1/√3) = 5π/6 (inverse cotangent of -1/√3)

Therefore, arcctg(ctg(2π/3)) simplifies to 5π/6.

6. ctg(x) = -0.5: To find x, we need to find the angle whose cotangent is -0.5. The standard angle where the cotangent is -0.5 is 7π/6. Adding the period of cotangent, we can generalize the solutions as: x = 7π/6 + πn, where n is an integer.

7. ctg(x) = 0: To find x, we need to find the angle whose cotangent is 0. The standard angle where the cotangent is 0 is π/2. Adding the period of cotangent, we can generalize the solutions as: x = π/2 + πn, where n is an integer.

Please note that the values of angles given in radians.