Chapter 5.5, Problem 55E

Solution

a)

To express: The height $h$ in terms of angle $A$ and length $AB$ .

  $h$ in terms of angle $A$ and length $AB$ is $h=AB\sin A$ .

Given information:

Consider the given triangle.

  

Calculation:

Express $h$ in terms of angle $A$ and length $AB$ .

  $\begin{array}{c}\sin A=\frac{h}{AB}\\ h=AB\sin A\end{array}$

Therefore, $h$ in terms of angle $A$ and length $AB$ is $h=AB\sin A$ .

b)

To state: The condition on length $BC$ that will result in a no triangle is formed.

The required condition is $BC<AB\sin A$ .

Calculation:

In order for the triangle not to exist, $BC<h$ .

Express $h$ in terms of angle $A$ and $AB$, and set the condition for the triangle not to exist.

  $BC<AB\sin A$

Therefore, the required condition is $BC<AB\sin A$ .

c)

To state: The condition on length $BC$ that will result in a unique triangle is formed.

The required condition is $BC=AB\sin A$ or $BC\ge AB$ .

Calculation:

In order to get only one triangle, $BC=h$ or $BC\ge AB$ .

Express $h$ in terms of angle $A$ and $AB$, and set the condition for the unique triangle to exist.

  $BC=AB\sin A$ or $BC\ge AB$

Therefore, the required condition is $BC=AB\sin A$ or $BC\ge AB$ .

d)

To state: The condition on length $BC$ that will result in two possible triangle is formed.

The required condition is $AB\sin A<BC<AB$

Calculation:

In order for the problem to have 2 solutions, $h<BC<AB$ .

Express $h$ in terms of angle $A$ and $AB$, and set the condition for the triangle to have two solutions.

  $AB\sin A<BC<AB$

Therefore, the required condition is $AB\sin A<BC<AB$ .