Question

the angles that share the same tangent value as tan(45°) have terminal sides in quadrant. options: ii, iii, iv, (none of the quadrants)

Explanation:

Step1: Recall tangent function property

The tangent function has a period of \(180^\circ\) (or \(\pi\) radians), and \(\tan(\theta)=\tan(\theta + 180^\circ n)\) for any integer \(n\). Also, \(\tan(45^\circ) = 1\), and the tangent function is positive in quadrants I and III (since \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and in quadrant III, both \(\sin\theta\) and \(\cos\theta\) are negative, so their ratio is positive).

Step2: Determine the quadrant

Since \(45^\circ\) is in quadrant I, angles with the same tangent value as \(\tan(45^\circ)\) will be \(45^\circ+ 180^\circ n\). For \(n = 1\), we get \(45^\circ+180^\circ=225^\circ\), which is in quadrant III. So the terminal sides of angles with the same tangent as \(\tan(45^\circ)\) lie in quadrant III (and also quadrant I, but among the given options, III is the correct one here).

Answer:

III (the option labeled "III" in the dropdown)