Question

How do you find the measurement of the included angle $\theta $ between two sides of the triangle if the two sides measurements are 27m and 46m and the area of the triangle is 252 square meters?

Answer 1

Hint:
Use the law of sine formula according to what we have to find and what we have been given in the question.

Complete step by step solution:
When we have the two sides of the triangle, and its total area, we have a formula from law of sines from which we can get the required angle and the required angle is the angle between the two given sides.
The formula is
$Area = \dfrac{1}{2} \times a \times b \times \sin C$
Where
a= measurement of side 1
b= measurement of side 2
C= The angle between the given side 1 and side 2.
Area= Area of the triangle.
This formula can be accordingly changed just not to find the area but also any sides and also finding the angle between the two given sides.

From the question, we can see that we are given two sides of the triangle and also the area of the triangle, which are
a =27 meters
b= 46 meters
we given angle = $\theta $
and lastly, the area of triangle =252 square meters
From the given values, we substitute them into the formula given above for find the area .

$252 = \dfrac{1}{2} \times 27 \times 46 \times \sin \theta $
Then we begin simplification such that we find the value of the angle at the end
$
  \sin \theta = \dfrac{{2 \times 252}}{{27 \times 46}} \\
  \sin \theta = \dfrac{{28}}{{129}} \\
  \sin \theta = 0.2171 \\
  \theta = 12.536 \\
$
In the end, we got the value of the angle by doing inverse trigonometry, by sending the sine to the right side, and finding its value.

Note:
To use the laws of sine formula, we have to make sure from the question that the given angle or the angle to be found is between the given sides whose measurements are given in the question and also we can leave the value at the inverse trigonometry based on the weightage of the question.