Can you make that wonderful observation?

Geometry Level 1

A triangle has sides of magnitude $1$ , $\sin x$ , and $\cos x$ .

where $0 < x < \frac { \pi }{ 2 } .$

Find the largest angle of the triangle in degrees.

Assume that the triangle is non-degenerate.

The answer is 90.

1 solution

Milind Prabhu
Nov 13, 2015

By careful observation we see that in the triangle ${ \sin ^{ 2 }{ x } +\cos ^{ 2 }{ x } =1 }$

By converse of the Pythagorean Theorem the angle opposite to the side of magnitude $1$ is a right angle. Since the given triangle is non-degenerate the other two angles are non zero. Since they sum up to a right angle both of them must be less than a right angle. Hence the greatest angle in the triangle is $\boxed { { 90 }^{ o } }$

What does non-degenerate mean?

Star Chou - 4 years, 6 months ago

Just like we refer to conic sections can degenerate to lines, its when figures can "collapse" to a lower dimension figure. Specifically if an angle of a triangle is 0, it degenerates to a line segment (with angles of 0°, 0°, and 180°).

Jerry McKenzie - 3 years, 5 months ago

In simple terms, it means that the triangle isn't a line segment or some other shape.

Shubhrajit Sadhukhan - 4 months, 2 weeks ago

According to the problem 0<x<\pi/2, but \pi/2 equals 90\degree

roberto paningbatan - 3 years, 8 months ago

In this problem, $x$ is not the largest angle. The largest angle is the one that is opposite the hypotenuse.

Tirthankar Mazumder - 9 months, 3 weeks ago

Can you make that wonderful observation?

The answer is 90.

1 solution

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