Question

20 in △abc, m∠c = 90. if ab = 5 and ac = 4, which statement is not true? 1) cosa = \frac{4}{5} 2) tana = \frac{3}{4} 3) sinb = \frac{4}{5} 4) tanb = \frac{5}{3}

Explanation:

Step1: Find the length of BC

Using the Pythagorean theorem \(AB^{2}=AC^{2}+BC^{2}\).
Given \(AB = 5\) and \(AC = 4\), then \(BC=\sqrt{AB^{2}-AC^{2}}=\sqrt{25 - 16}=\sqrt{9}=3\).

Step2: Calculate \(\cos A\)

\(\cos A=\frac{AC}{AB}\) (adjacent side over hypotenuse in right - triangle trigonometry).
Since \(AC = 4\) and \(AB = 5\), \(\cos A=\frac{4}{5}\).

Step3: Calculate \(\tan A\)

\(\tan A=\frac{BC}{AC}\) (opposite side over adjacent side).
Since \(BC = 3\) and \(AC = 4\), \(\tan A=\frac{3}{4}\).

Step4: Calculate \(\sin B\)

\(\sin B=\frac{AC}{AB}\) (opposite side over hypotenuse for angle \(B\)).
Since \(AC = 4\) and \(AB = 5\), \(\sin B=\frac{4}{5}\).

Step5: Calculate \(\tan B\)

\(\tan B=\frac{AC}{BC}\) (opposite side over adjacent side for angle \(B\)).
Since \(AC = 4\) and \(BC = 3\), \(\tan B=\frac{4}{3}\).

Answer:

1. \(\tan B=\frac{5}{3}\)