Question

question 28 (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) 6.2 cm
b) 5.4 cm
c) 8.0 cm
d) 7.3 cm

Explanation:

Step1: Find adjacent side \(b\)

In a right - triangle, \(\cos A=\frac{b}{c}\), where \(c = 5.0\mathrm{cm}\) (hypotenuse) and \(\cos A=0.75\).
So, \(b = c\times\cos A=5\times0.75 = 3.75\mathrm{cm}\)

Step2: Find opposite side \(a\)

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

Step3: Calculate the area of the triangle

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

Answer:

a) \(6.2\mathrm{cm}\)