To find the value of sin x based on the given values of tan x and cos x, we can use the definition of tangent and the Pythagorean identity.
First, recall that:
tan x = sin x / cos x
We are given:
tan x = 3 and cos x = 1/2
Using the tangent definition, we can write:
3 = sin x / (1/2)
This implies:
sin x = 3 * (1/2) = 3/2
However, sin x cannot be greater than 1 (since the sine of any angle must be between -1 and 1), so we need to verify our calculations.
We can check using the Pythagorean identity:
sin² x + cos² x = 1
We know cos x = 1/2, so:
sin² x + (1/2)² = 1
sin² x + 1/4 = 1
sin² x = 1 – 1/4
sin² x = 3/4
Therefore:
sin x = ±√(3/4) = ±√3 / 2
Thus, the value of sin x can either be √3/2 or -√3/2, depending on the quadrant in which x lies.