find the area of the triangle below. carry your intermediate computations to at least four decimal places…

Explanation:

Step1: Recall area - formula for a triangle with two - sides and included - angle

The formula for the area of a triangle with two sides (a) and (b) and the included - angle (\theta) is (A=\frac{1}{2}ab\sin\theta). Here, (a = 5\mathrm{cm}), (b = 3\mathrm{cm}), and (\theta=38^{\circ}).

Step2: Calculate the sine of the angle

We know that (\sin(38^{\circ})\approx0.6157) (using a calculator and keeping four decimal places).

Step3: Substitute values into the formula

Substitute (a = 5), (b = 3), and (\sin(38^{\circ})\approx0.6157) into the formula (A=\frac{1}{2}ab\sin\theta). So (A=\frac{1}{2}\times5\times3\times0.6157).

Step4: Perform the multiplication

First, (\frac{1}{2}\times5\times3=\frac{15}{2}=7.5). Then (A = 7.5\times0.6157=4.61775).

Step5: Round the answer

Rounding (4.61775) to the nearest hundredth, we get (A\approx4.62\mathrm{cm}^{2}).

Answer:

(4.62\mathrm{cm}^{2})