Question

question 8 (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) 8.0 cm
c) 7.3 cm
d) 5.4 cm

Explanation:

Step1: Find adjacent side \(b\)

In a right - triangle, \(\cos A=\frac{b}{c}\), where \(c\) is the hypotenuse and \(b\) is the adjacent side. Given \(c = 5.0\) cm and \(\cos A=0.75\), then \(b = c\times\cos A\).

$$b=5\times0.75 = 3.75$$

Step2: Find opposite side \(a\)

Using the Pythagorean theorem \(a=\sqrt{c^{2}-b^{2}}\). Substitute \(c = 5\) and \(b = 3.75\)

$$a=\sqrt{5^{2}-3.75^{2}}=\sqrt{25 - 14.0625}=\sqrt{10.9375}\approx3.3$$

Step3: Calculate the area of the triangle

The area of a right - triangle \(S=\frac{1}{2}ab\). Substitute \(a\approx3.3\) and \(b = 3.75\)

$$S=\frac{1}{2}\times3.3\times3.75=\frac{1}{2}\times12.375 = 6.2$$

Answer:

a) \(6.2\) cm