Final answer:
The equation 3cotx + √3 = 0 has solutions at x = 5π/6 and x = 11π/6 for the interval 0 ≤ x < 2π. To simplify arccos(sin 50°), we apply the complementary angle identity, obtaining a result of 40°.
Step-by-step explanation:
To solve the equation 3cotx + √3 = 0, we first isolate cotx by subtracting √3 from both sides, giving us 3cotx = -√3. Dividing both sides by 3, we get cotx = -√3/3. Now, we use the inverse cotangent function, arccot(x), to find x. Since cotangent is the reciprocal of tangent, we are looking for an angle whose tangent is -1/√3. This is 5π/6 or 150° and 11π/6 or 330° within the interval 0 ≤ x < 2π.
For simplify arccos(sin 50°), we can use the complementary angle identity since sine and cosine are co-functions. The sine of an angle is the same as the cosine of its complement. Therefore, sin 50° is equivalent to cos 40°. Because arccos is the inverse function of cosine, arccos(cos 40°) returns 40°. So, simplify arccos(sin 50°) equals 40°.