To find the measure of $\angle B$ in a triangle $\triangle ABC$, the equation used will depend on the sides and angles given. Generally, trigonometric identities or the Law of Sines and Law of Cosines are used. Here are two common approaches:

Using the Law of Sines

The Law of Sines states: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ where $a, b,$ and $c$ are the lengths of the sides opposite to angles $A, B,$ and $C$ respectively.

If you have the lengths of sides $a$ and $b$ and the measure of angle $A$, you can find $\angle B$ using: $\sin(B) = \frac{b \sin(A)}{a}$

Using the Law of Cosines

The Law of Cosines is given by: $c^2 = a^2 + b^2 - 2ab \cos(C)$

If you have the lengths of sides $a, b,$ and $c$, you can find $\angle B$ using: $b^2 = a^2 + c^2 - 2ac \cos(B)$ Solving for $\cos(B)$: $\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}$

So the equations you might use are:

1. $\sin(B) = \frac{b \sin(A)}{a}$
2. $\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}$

To summarize:

1. $\sin(B) = \frac{b \sin(A)}{a}$
2. $\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}$

Would you like more details or have any questions? Here are five other math questions you might consider:

1. How do you use the Law of Cosines to find the length of a side in a triangle?
2. What are the properties of similar triangles?
3. How do you find the area of a triangle using trigonometry?
4. How can you determine if a triangle is right, acute, or obtuse using its sides?
5. What is the relationship between the angles in a triangle?

Tip: When solving for angles in a triangle, always ensure you have the correct sides and angles to use the appropriate trigonometric law or identity.