Right triangle definitions of trig functions and related applications

Some of the most fundamental trigonometric identities are those derived from the Pythagorean Theorem. These are defined using a right triangle:

right triangle

At this stage in the course the Pythagorean Theorem should be second nature to you. If you have not yet learned it, learn it now.

$a^{2}+b^{2}=c^{2}$

$a$ and $b$ of course are the legs or the adjacent and opposite edges, and $c$ is the hypotenuse, the longest side, the side that does not include the right angle. This formula only works for a right angle triangle. If the angle shown as a right angle in the diagram were obtuse, larger than a right angle, then $c^{2}$ would be larger than the Pythagorean sum. If the angle shown as a right angle were smaller than a right angle, then $c^{2}$ would be smaller than $a^{2}+b^{2}$ .

$\cos \theta ={\frac {x}{r}}={\frac {x}{1}}=x$	$\sec \theta ={\frac {r}{x}}={\frac {1}{x}}$
$\sin \theta ={\frac {y}{r}}={\frac {y}{1}}=y$	$\csc \theta ={\frac {r}{y}}={\frac {1}{y}}$
$\tan \theta ={\frac {y}{x}}$	$\cot \theta ={\frac {x}{y}}$

Resources

• Right triangle definitions of trig functions and related applications. Written notes created by Professor Esparza, UTSA.