Question

a =
round your answer to the nearest hundredth
(right triangle with vertices a, b, c; right angle at c)

Explanation:

Step1: Identify triangle type

The triangle is right - angled at \(C\), with \(AC = 4\), \(AB=5\). Let \(BC=x\). By Pythagorean theorem, \(AB^{2}=AC^{2}+BC^{2}\).

Step2: Apply Pythagorean theorem

We know that \(AB = 5\), \(AC = 4\). So \(5^{2}=4^{2}+x^{2}\), which is \(25 = 16+x^{2}\). Then \(x^{2}=25 - 16=9\), so \(x = 3\) (since length can't be negative).

Step3: Calculate angle \(A\)

We can use trigonometric ratios. \(\sin A=\frac{BC}{AB}=\frac{3}{5}\), \(\cos A=\frac{AC}{AB}=\frac{4}{5}\), \(\tan A=\frac{BC}{AC}=\frac{3}{4}\). To find angle \(A\), we can use \(\sin^{- 1}(\frac{3}{5})\) or \(\cos^{-1}(\frac{4}{5})\) or \(\tan^{-1}(\frac{3}{4})\).
Using \(\sin^{-1}(\frac{3}{5})\): \(\sin^{-1}(0.6)\approx36.87^{\circ}\), which rounds to \(37^{\circ}\) (to the nearest whole number).

Answer:

\(37^{\circ}\)