Final answer:
The measure of the included obtuse angle can be found by using the area formula for a triangle with two sides and an included angle, rearranging for the sine of the angle, taking the inverse sine, and making adjustments if necessary, rounding to the nearest tenth of a degree.
Step-by-step explanation:
To find the measure of the included angle of a triangle with a given area and side lengths, we can use the formula for the area of a triangle when two sides and the included angle are known, which is Area = (1/2) × side A × side B × sin(included angle). Given the area of the triangle is 2034 square units and the two sides are 74 and 69 units, we can set up the equation:
2034 = (1/2) × 74 × 69 × sin(included angle)
To solve for the included angle, we rearrange the equation:
sin(included angle) = 2034 × 2 / (74 × 69)
We then take the inverse sine (arcsin) of both sides to find the angle:
included angle = arcsin(2034 × 2 / (74 × 69))
Because we are looking for an obtuse angle, which is greater than 90 degrees but less than 180 degrees, we subtract the result from 180 degrees if necessary, as the arcsin function will return an acute angle.
Finally, we round the answer to the nearest tenth of a degree to find the measure of the included obtuse angle.