Final answer:
To solve the equation '6 cos x - 5 sin x = 0', first express it in terms of one trigonometric function '6 = 5 tan x' and then find x such that 'tan x = 6/5' within the interval 0 < x < 2π.
Step-by-step explanation:
To solve the trigonometric equation 6 cos x - 5 sin x = 0, we start by isolating one of the trigonometric functions:
6 cos x = 5 sin x
Divide both sides by cos x (assuming cos x is not zero):
6 = 5 tan x
Now divide by 5:
tan x = ±6/5
Find the angle whose tangent is 6/5. This can be done using a calculator or trigonometric tables.
The solutions in the interval 0 < x < 2π would be the angles where the tangent has the value of 6/5, as well as the angles in the third or fourth quadrant where the tangent also equals 6/5.
Remember that tan x is positive in the first and third quadrants, so you must check for solutions in those quadrants.