Expert: Scotto - 7/9/2009

Question
1. Given ΔABC with <B = 60° , b = 21.5 and c = 12.4, find
<C , to the nearest tenth of a degree.

2. Find the area of ΔABC given a = 30 cm, b = 18 cm, and
c = 24 cm.


3. In ΔABC, a = 6, b = 10, and c = 7.  Find the measure of the largest angle to the nearest tenth of a degree.

4. Find the area of ΔABC if  a = 8, b = 15, and <C = 40°.


5.Find the area of the triangle ΔABC If a=37cm, b=25cm and c=42cm.


Answer
1. The law of cosines can be applied.
It is a² = b² + c² - 2bc cos C,
so all that has te be done is to find c.

That is, 2bc cos C = b² + a² - c²,
so cos C = (b² + a² - c²)/(2bc),
so C = arccos(b² + a² - c²)/(2bc).


2. The area A = bh/2 where b is the base and h is the height.

Using the law of cosines, an angle can be found.
Let one side next to this angle be the base.
Let the other side be the hypoteneuse on the right triangle
so that one of the sides is the height.  Using this angle,
the height of the triangle can be found.
Once the height h is found and the base b, find the area A.


3. The largest angle would be the one oppositive the largest side.
Since b is the longest side, the angle is B.

The law of cosines for this case a² + c² - 2ac cos B = b².
From here, it can be said that cos B = (a² + c² - b²)/(2ac).

Note that the cos function is 1-to-1 between 0 and 180 degrees,
and that's where the angle on the triangle is found.


4. Take the side a as the base.  The height h can be found since
h/b = cos C and b and C are known.  Using this, the area A = ah/2.


5. Find the area of the triangle ΔABC If a=37cm, b=25cm and c=42cm.
Using the law of cosines, we can find one of the angles.
Using this angle and a side next to it as the base b, the height h can be found.  The area is bh/2.