Sin and Cos relationship with Triangle sides

Question

In a triangle ABC, ${sinA < \frac{a}{c}}$ and ${cosA > \frac{b}{c}}$. Which of the statements below are always false regarding triangle ABC?

1. ABC is an acute triangle
2. ABC is an isosceles triangle
3. ABC is an obtuse triangle where B is an obtuse angle
4. ABC is a right triangle where A is a right angle
5. ABC is an obtuse triangle where A is the obtuse angle

I only understand that the fourth one is false because the relationship would be equal instead of < or >, but how would the rest be solved? Any help would be greatly appreciated!

Answer 1

I suggest you draw the possibilities. I've drawn a few of them for you.

Since you already know that the given triangle is not a right triangle, the leftmost drawing should be easy to understand.

Look at the middle drawing. Since angle $A$ has not changed, $\sin A$ and $\cos A$ are also unchanged. But $a$ has become longer, and $b$ has become longer (and $c$ is unchanged). Therefore, $\sin A<\frac ac$ and $\cos A<\frac bc$.

Look at the rightmost drawing. Since angle $A$ has not changed, $\sin A$ and $\cos A$ are also unchanged. But $a$ has become longer, and $b$ has become shorter (and $c$ is unchanged). Therefore, $\sin A<\frac ac$ and $\cos A>\frac bc$.

I hope this gives you enough hints to answer your questions.