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trig quiz
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the hypotenuse, c, of right \\( \triangle a b c \\) 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 \\( \mathrm{cm}^{2} \\) ?
\\( \bigcirc \\) a) \\( 5.4 \mathrm{~cm} \\)
\\( \bigcirc \\) b) \\( 6.2 \mathrm{~cm} \\)
\\( \bigcirc \\) c) \\( 8.0 \mathrm{~cm} \\)
\\( \bigcirc \\) d) \\( 7.3 \mathrm{~cm} \\)
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Explanation:

Step1: Find the adjacent side to angle \(A\)

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

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

Step2: Find the opposite side to angle \(A\)

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

$$a=\sqrt{5^{2}-3.75^{2}}=\sqrt{25 - 14.0625}=\sqrt{10.9375}\approx3.31$$

Step3: Calculate the area of the right - triangle

The area of a right - triangle \(A=\frac{1}{2}ab\). Substitute \(a\approx3.31\) and \(b = 3.75\)

$$A=\frac{1}{2}\times3.31\times3.75=\frac{1}{2}\times12.4125 = 6.20625\approx6.2$$

Answer:

B. \(6.2\mathrm{cm}^{2}\)