Answer:
x = arctan(1/2)+ k(π) or x ≈ 0.4636 + k(π), where k is any integer.
Step-by-step explanation:
The equation [tex] \sf \tan x = \frac{1}{2} [/tex] means we're looking for the angle [tex] \sf x [/tex] whose tangent is[tex] \sf \frac{1}{2} [/tex]. The function [tex] \sf \tan x [/tex] is periodic with a period of [tex] \sf \pi [/tex], meaning it repeats every [tex] \sf \pi [/tex] radians.
1. First, find the principal value using the arctan (inverse tangent) function:
[tex] \sf \rightarrow x = \arctan\left(\frac{1}{2}\right)[/tex]
- This gives the angle in the first or fourth quadrant where tangent has the value [tex] \sf \frac{1}{2} [/tex].
2. Since the tangent function is periodic, the general solution includes all angles of the form:
[tex] \sf \rightarrow x = \arctan\left(\frac{1}{2}\right) + k\pi[/tex]
- where [tex] \sf k [/tex] is any integer, to account for the repeating nature of the tangent function across all quadrants.
3. Calculating the arctan of [tex] \sf \frac{1}{2} [/tex] (through a calculator for simplicity):
[tex] \sf \rightarrow \arctan\left(\frac{1}{2}\right) \approx 0.4636 \text{ radians} [/tex]
Note:
- Remember that the tangent function has a period of [tex] \sf \pi [/tex], so always include [tex] \sf+ k\pi [/tex] in the general solution for equations involving the tangent.
- To find the angle in degrees, if needed, convert radians by multiplying by [tex] \sf \frac{180}{\pi} [/tex].
- Use a calculator for more precise or decimal values when dealing with trigonometric functions like arctan, especially in test settings.
