Question

Two friends are writing practice problems to study for a trigonometry test. Sam writes the following problem for his friend Anna to solve:

In right triangle ABC, the measure of angle C is 90 degrees, and the length of side c is 8 inches.
Solve the triangle.
Anna tells Sam that the triangle cannot be solved. Sam says that she is wrong.
Who is right? Explain your thinking

Answer 1

Anna is right in her meaning concerning on triangle solvability.

Explanation:

The side
`c` represents the hypotenuse of a right triangle as
`C = 90^(\circ)` and is opposite to that angle. There are two ways to solve this triangle trigonometrically:

i) Law of Sine

`(a)/(\sin A) = (b)/(\sin B) = (c)/(\sin C)` (1)

ii) Law of Cosine

`c^(2) = a^(2) + b^(2) - 2\cdot a\cdot b \cdot \cos C` (2)

The Pythagorean Theorem is a particular case of the Law of Cosine for
`C = 90^(\circ)`

The triangle cannot be solved as there is an input missing, either another side or another angle. If
`C = 90^(\circ)`, then (2) is reduced into this form:

`c^(2) = a^(2)+b^(2)` (2b)

In this case we need to know the measure of either
`a` or
`b` to determine its counterpart and the values of the missing angles by (1). In nutshell, Anna is right.