Solving a triangle, given two sides and the measure of the included angle

Question

Let say you have a triangle

Angle A = 41 degrees , side b = 3.41 and c = 5.83

can you use pythagoras theorem to find the side a? and how can you find Angle B and C

Answer 1

There isn't a simple way to use the Pythagorean Theorem to find any of the unknown information about the triangle. It is possible to use some right triangle trigonometry in combination with the Pythagorean Theorem to find the unknown information, but it's simpler to use the Law of Cosines—to find the unknown side and then to find the unknown angles.

Answer 2

To find side $a$ use Law Of Cosines:

$$a^2=b^2+c^2-2\cdot b \cdot c \cos \alpha$$

To find angles $\beta$ and $\gamma$ use Sine Law :

$$\frac{a}{\sin \alpha} =\frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$$