Questions: Find the area K of the triangle. a=8, c=7, B=95° K= square units (Round to two decimal places as needed.)

Transcript text: Find the area K of the triangle. \[ a=8, c=7, B=95^{\circ} \] $\mathrm{K}=$ $\square$ square units (Round to two decimal places as needed.)

Solution

Solution Steps

To find the area $ K $ of the triangle given sides $ a $ and $ c $ and angle $ B $, we can use the formula: \[ K = \frac{1}{2} \times a \times c \times \sin(B) \] We will use Python to calculate this, ensuring to convert the angle $ B $ from degrees to radians before using the sine function.

Step 1: Given Values

We are given the following values for the triangle:

$ a = 8 $
$ c = 7 $
$ B = 95^\circ $

Step 2: Convert Angle to Radians

To use the sine function, we convert the angle $ B $ from degrees to radians: \[ B_{\text{radians}} = \frac{95 \times \pi}{180} \approx 1.6581 \]

Step 3: Calculate the Area

Using the formula for the area $ K $ of the triangle: \[ K = \frac{1}{2} \times a \times c \times \sin(B) \] Substituting the values: \[ K = \frac{1}{2} \times 8 \times 7 \times \sin(1.6581) \approx 27.8935 \]

Step 4: Round the Area

Rounding the area to two decimal places, we find: \[ K_{\text{rounded}} \approx 27.89 \]