Final answer:
To find solutions to the equation 6tan(\(\alpha\)) + 7 = 4 in the given interval, we simplify to tan(\(\alpha\)) = -0.5 and use arctangent, taking into account the periodic nature of the tangent function and finding both angles within the interval 0 to 2\(\pi\) radians.
Step-by-step explanation:
The student has asked us to find all solutions to the equation 6tan(\(\alpha\)) + 7 = 4, with the restriction that 0 \(\leq\alpha\leq\) 2\(\pi\). To solve for \(\alpha\), we first simplify the equation to tan(\(\alpha\)) = -0.5 by subtracting 7 from both sides and then dividing by 6.
We then use the arctangent function to find the angle that gives the tangent of -0.5. Because there are two angles between 0 and 2\(\pi\) that have the same tangent value (one in the second quadrant and one in the fourth quadrant), we need to find both of these angles.
The principal value of arctan(-0.5) gives us an angle in the fourth quadrant. To find the angle in the second quadrant, we add \(\pi\) to this value. Remember to check your answer and ensure it makes sense within the given parameters.