Question

Solve the following trigonometric inequalities for 0≤x<2π:

a) tan(x)>√(3)
​b) cos(x)> √(3)/2
​c) 3tan(x)−√(3)<0
d) No solution in the given range

Answer 1

To solve the trigonometric inequalities, tan(x) must be located in intervals where the tangent function is greater than √(3), cos(x) must be in the intervals where the cosine function exceeds √(3)/2, and 3tan(x) - √(3) must be negative, which can be found by reviewing the unit circle and the graphs of these functions.

Step-by-step explanation:

To solve the given trigonometric inequalities for 0≤x<2π, we first need to consider the function and where it satisfies the inequality within one period. Here's how we can approach each:

1. tan(x)>√(3): Since tan(x) equals √(3) at π/3 and 4π/3, look at the sections of the unit circle where tan(x) is greater than √(3), which is between (π/3, 2π/3) and (4π/3, 5π/3).
2. cos(x)> √(3)/2: For cosine to be greater than √(3)/2, the angle x must be within the first quadrant around 0 and 60 degrees, or in radians, between (0, π/6) and (11π/6, 2π).
3. 3tan(x)−√(3)<0: Divide by 3 to get tan(x) < √(3)/3. Analyze the tan graph to find the interval where this inequality holds, which will be from (π/3, π) and (4π/3, 2π).
4. No solution would apply if an inequality is inherently contradictory or never true for the given range, which is not the case here.