Question

question 5 (1 point)
the hypotenuse, c, of right △abc is 5.0 cm long. given the trigonometric ratio
cos a = 0.75 for angle a, what is the area of the triangle to the nearest tenth of a cm²?

a) 5.4 cm
b) 8.0 cm
c) 6.2 cm
d) 7.3 cm

Explanation:

Step1: Find adjacent side

In a right - triangle, \(\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}\). Given \(c = 5.0\) cm and \(\cos A=0.75\), let the adjacent side to angle \(A\) be \(b\). Then \(b = c\times\cos A=5\times0.75 = 3.75\) cm.

Step2: Find opposite side

Using the Pythagorean theorem \(a=\sqrt{c^{2}-b^{2}}\). Substitute \(c = 5\) and \(b = 3.75\) into the formula: \(a=\sqrt{5^{2}-3.75^{2}}=\sqrt{25 - 14.0625}=\sqrt{10.9375}\approx3.3\) cm.

Step3: Calculate area

The area of a right - triangle is \(S=\frac{1}{2}ab\). Substitute \(a\approx3.3\) and \(b = 3.75\) into the formula: \(S=\frac{1}{2}\times3.3\times3.75=\frac{1}{2}\times12.375 = 6.1875\approx6.2\) \(cm^{2}\).

Answer:

C. \(6.2\) \(cm^{2}\)