Solve for b.
$b=[?]$
Law of Sines: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Answer (1)
The Law of Sines states: s i n A a = s i n B b = s i n C c .
To solve for b , set up a proportion using another known ratio.
If a , A , and B are known: b = s i n A a s i n B .
If c , C , and B are known: b = s i n C c s i n B .
b = sin A a sin B or b = sin C c sin B
Explanation
Understanding the Law of Sines The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. We are given the Law of Sines: sin A a = sin B b = sin C c Our goal is to solve for b .
Isolating b To isolate b , we can set up an equation using any of the other ratios in the Law of Sines. For example, if we know a and A , we can use the ratio s i n A a . If we know c and C , we can use the ratio s i n C c .
Solving for b Let's assume we know a , A , and B . Then we can write: sin A a = sin B b To solve for b , we multiply both sides of the equation by sin B : b = sin A a sin B Alternatively, if we know c , C , and B , we can write: sin C c = sin B b To solve for b , we multiply both sides of the equation by sin B : b = sin C c sin B
Final Answer Therefore, depending on the known values, b can be expressed as: b = sin A a sin B or b = sin C c sin B
Examples
The Law of Sines is incredibly useful in surveying and navigation. Imagine you're a surveyor trying to determine the distance to a point across a river. You can measure the angles to that point from two known locations on your side of the river and the distance between those locations. Using the Law of Sines, you can then calculate the distance to the point across the river without ever crossing it! This principle is also used in GPS technology to calculate distances based on angles and known positions of satellites.
