SOLUTION: A triangular piece of wood has side lengths 14 cm, 19 cm and 30 cm. The angle between the 14 cm and 19 cm sides need to be known to see if it will fit properly into a cabinet. What

Algebra ->  Trigonometry-basics -> SOLUTION: A triangular piece of wood has side lengths 14 cm, 19 cm and 30 cm. The angle between the 14 cm and 19 cm sides need to be known to see if it will fit properly into a cabinet. What      Log On


   

Question 1181689: A triangular piece of wood has side lengths 14 cm, 19 cm and 30 cm. The angle between the 14 cm and 19 cm sides need to be known to see if it will fit properly into a cabinet. What is the angle, accurate to one decimal place, between the 14 cm and 19 cm sides?
Answer by ikleyn(52852) About Me  (Show Source):
You can put this solution on YOUR website!
.


            There are 2 (two) ways to solve the problem.

            One way is to use the cosine law.

            Another way is via the area.

            I will show you this second way. 


Calculate the area of the triangle, using the Heron's formula.


Doing this way, you get for the area of the triangle the value of 101.666  cm^2.


The area of the triangle also can be found using the formula


    area = %281%2F2%29%2A14%2A19%2Asin%28alpha%29,


where alpha  is the angle between the sides  of  14 and 19 cm.


So,  %281%2F2%29%2A14%2A19%2Asin%28alpha%29 = 101.666 


It gives  sin%28alpha%29 = %282%2A101.666%29%2F%2814%2A19%29 = 0.764.


Also, notice that  14^2 + 19^2 = 557 < 900 = 30^2.


Hence, the angle alpha must be acute.


It gives for alpha  the unique answer  


    alpha = arcsin(0.764) = 0.86949 radians = 49.8 degrees   (rounded as requested).

Solved.

---------------

So, having two ways to express / (to calculate) the area, we obtain the equation to find the sine of the angle.

From equation, we determine the sine of the angle and then the angle itself.