Question

what is the reciprocal of tanb in the triangle below?
a
b c
tanc
tana
tan^{-1}c
tan^{-1}a

Explanation:

Step1: Recall tangent in right triangle

In right triangle \(ABC\) with right angle at \(C\), \(\tan B=\frac{\text{opposite}}{\text{adjacent}}=\frac{AC}{BC}\).

Step2: Find \(\tan A\)

\(\tan A = \frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AC}\).

Step3: Find reciprocal of \(\tan B\)

The reciprocal of \(\tan B=\frac{AC}{BC}\) is \(\frac{BC}{AC}\), which is \(\tan A\). Wait, no, wait. Wait, angle \(C\) is right angle, so angles \(A\) and \(B\) are complementary. Wait, \(\tan B=\frac{AC}{BC}\), reciprocal is \(\frac{BC}{AC}\), and \(\tan A=\frac{BC}{AC}\)? Wait, no, angle at \(A\): opposite is \(BC\), adjacent is \(AC\), so \(\tan A = \frac{BC}{AC}\). And \(\tan B=\frac{AC}{BC}\), so reciprocal of \(\tan B\) is \(\frac{BC}{AC}=\tan A\)? Wait, no, wait the options: wait, the triangle has right angle at \(C\), so angle \(B\) and angle \(A\) are acute. Wait, \(\tan B=\frac{AC}{BC}\), reciprocal is \(\frac{BC}{AC}\), which is \(\tan A\)? Wait, no, \(\tan A=\frac{BC}{AC}\), so reciprocal of \(\tan B\) is \(\tan A\)? Wait, but the options have \(\tan A\) as an option? Wait, no, the options are \(\tan C\), \(\tan A\), \(\tan^{-1}C\), \(\tan^{-1}A\). Wait, angle \(C\) is \(90^\circ\), \(\tan C\) is undefined. Wait, maybe I made a mistake. Wait, in right triangle, \(\tan B=\frac{AC}{BC}\), reciprocal is \(\frac{BC}{AC}\), and \(\tan A=\frac{BC}{AC}\), so reciprocal of \(\tan B\) is \(\tan A\)? Wait, no, \(\tan A\) is \(\frac{BC}{AC}\), which is the reciprocal of \(\tan B=\frac{AC}{BC}\). So the reciprocal of \(\tan B\) is \(\tan A\)? Wait, but let's check again. Wait, angle \(B\): opposite side is \(AC\), adjacent is \(BC\). Angle \(A\): opposite side is \(BC\), adjacent is \(AC\). So \(\tan B = \frac{AC}{BC}\), \(\tan A=\frac{BC}{AC}\), so \(\frac{1}{\tan B}=\frac{BC}{AC}=\tan A\). Wait, but the options: one of the options is \(\tan A\). Wait, but the options are: \(\tan C\) (undefined, since \(C = 90^\circ\), \(\tan 90^\circ\) is undefined), \(\tan A\), \(\tan^{-1}C\) (arctan of \(C\), but \(C\) is \(90^\circ\), arctan(90°) is not standard), \(\tan^{-1}A\) (arctan of angle \(A\), not relevant). Wait, maybe I messed up. Wait, no, the reciprocal of \(\tan B\) is \(\cot B\), and \(\cot B=\tan(90^\circ - B)=\tan A\), since \(A + B=90^\circ\). So \(\cot B=\tan A\), which is the reciprocal of \(\tan B\) (since \(\cot B=\frac{1}{\tan B}\)). So the reciprocal of \(\tan B\) is \(\tan A\). Wait, but the options have \(\tan A\) as an option. Wait, the options are:

• \(\tan C\): undefined (since \(C = 90^\circ\), \(\tan 90^\circ\) is undefined)
• \(\tan A\): \(\tan A=\frac{BC}{AC}\), which is \(\frac{1}{\tan B}\) (since \(\tan B=\frac{AC}{BC}\))
• \(\tan^{-1}C\): arctan of \(90^\circ\), not a valid trigonometric value in the context of triangle angles (arctan takes a ratio, not an angle measure here)
• \(\tan^{-1}A\): arctan of angle \(A\), not relevant

So the correct answer should be \(\tan A\), which is option B (assuming the options are labeled, but in the given options, the second option is \(\tan A\)). Wait, the user's options: the second option is \(\tan A\). So the reciprocal of \(\tan B\) is \(\tan A\).

Wait, no, wait I think I made a mistake. Wait, \(\tan B = \frac{AC}{BC}\), reciprocal is \(\frac{BC}{AC}\), and \(\tan A = \frac{BC}{AC}\), yes. So reciprocal of \(\tan B\) is \(\tan A\).

Answer:

\(\tan A\) (the second option, assuming the options are: 1. \(\tan C\), 2. \(\tan A\), 3. \(\tan^{-1}C\), 4. \(\tan^{-1}A\)) So the answer is the option with \(\tan A\), i.e., the second option (if labeled as B, then B. \(\tan A\)).