If I know the length of two sides of an acute or obtuse triangle and know the angle between these two sides, how can I get one of the missing two angles?
(Obviously, the final missing angle = 180 degrees - the known angle - one of the missing angles.)
Triangle is points $A, B, C$.
Known sides: $\overline{AB}$ & $\overline{BC}$.
Known angle: $\angle ABC$.
On
If the angle in between sides is known use first cosine rule, then sine rule.
If the angle in between sides is not known, but opposite to one of them only, then we have $two$ solutions because of the ambiguity
Draw a ray with known side and known angle. Cut the ray twice with known side as radius of circle.
On
If the angle you know is facing one of the sides you know, then it's best to use the Sine Rule. If the angle is between the two sides you know, then you have to use the Cosine Rule first to find the third side. From there you can continue with either the Sine or Cosine Rule.
Let's say you have a triangle with sides $a$, $b$, and $c$ and angles $A$, $B$, and $C$ (Side $a$ faces angle $A$, side $b$ faces angle $B$ and side $c$ faces angle $C$)...
The Law of Sines (also called Sine Rule) states that: $$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$$
If you substitute the values that you know, you can find the missing angles by cross multiplication.
The Law of Cosines (also called Cosine Rule) states that: $$c^2 = a^2 + b^2 − 2ab\cos(C)$$ ; where angle $C$ is facing side $c$ and is between the two sides $a$ and $b$.
Here if you substitute the values you know, you can find the third side $c$ facing the angle $C$. And then you can use the Sine or Cosine rule to find the missing angles.
NOTE that these examples are general as I do not know the details of your question. So try and understand the rules, and from there decide which you can use to find your answer.
Hint: Cosine rule:
Let us say that the sides of a triangle have a length $a,b,c$. And $\alpha$ is the opposite angle of the $A$, then: $$a^2=b^2+c^2-2bc\cos(\alpha).$$