# Trigonometric Ratios and Converse of Pythagoras Theorem Lemma

I am not quite sure what the question is asking here really. I understand the Lemma. Do I prove that $$a^2+b^2=1$$, using the given expressions for $$a$$ and $$b$$?

Thank you!

• Please do not use pictures. – Dietrich Burde Jun 12 at 15:02
• What is wrong with using a picture? Please explain. I did not see it anywhere in the rules that pictures are not permitted. – PomPom Jun 12 at 15:07
• See here for some good arguments not to use pictures. – Dietrich Burde Jun 12 at 15:19
• Thank you, fair comment! – PomPom Jun 12 at 15:23

Yes, you need to prove that $$a^2+b^2=1$$, using the given expressions for $$a$$ and $$b$$. You also need to show that $$a$$ and $$b$$ are positive (you'll see then why you need $$\alpha < 45^\circ$$).
• I am not quite sure why do I have to show that $a$ and $b$ are positive. I thought because I end up having $a=\cos2\alpha$ and $b=\sin2\alpha$ It is to show that $2\alpha$ is an acute angle. – PomPom Jun 12 at 16:25
• @PomPom The converse of the Lemma is saying "If $a,b$ are some pair of positive numbers and ...". And the point of the problem is that you can show that there exists some $\theta$ such that ..... without using trigonometry. The entire goal of this section/chapter seems to be to introduce $\cos$ and $\sin$ and their basic properties.. This means that you also need to avoid using more deep results. – N. S. Jun 12 at 19:03