High school trigonometry for physics

Trigonometric Functions

A right-angled triangle where c1 and c2 are the catheti and h is the hypotenuse.

The following trigonometric functions are defined for all right triangles. A right triangle is a triangle in which one of the angles is right, in other words 90 degrees. The sum of the two other angles always adds up to 90 degrees. The longest side in the triangle is called the hypotenuse, and it is always opposite to the right angle. The two remaining sides are called catheti. A cathetus is either one of the two sides which are adjacent to the right angle.

To make use of the trigonometric functions listed below, one has to understand that each and every one of them is always performed on a desired angle. One can imagine "standing" in the desired angle. The adjacent cathetus is the side directly connected with the angle, and the opposite cathetus is the side furthest away (not connected to the specific angle in any way). The dilemma arises when one is standing in the right angle, since then both catheti are adjacent and neither is opposite. To understand, for instance, why the cosine for 90 degrees is defined, it is necessary to understand the basics of the unit circle. For a high-school physicist, however, it is often more important to get the raw data, in this case the length of a side or the size of a specific angle, than understanding the mathematics behind it.

${\displaystyle \sin \theta ={\mathrm {opposite} \over \mathrm {hypoteneuse} }}$

${\displaystyle \cos \theta ={\mathrm {adjacent} \over \mathrm {hypoteneuse} }}$

${\displaystyle \tan \theta ={\mathrm {opposite} \over \mathrm {adjacent} }}$

${\displaystyle \cot \theta ={\frac {1}{\tan \theta }}={\mathrm {adjacent} \over \mathrm {opposite} }}$

A common mnemonic device for remembering these operations is Soh-Cah-Toa. Sin=Opposite/Hypoteneuse, Cos=Adjacent/Hypoteneuse, Toa=Opposite/Adjacent.

${\displaystyle \sec \theta ={\frac {1}{\cos \theta }}={\mathrm {hypoteneuse} \over \mathrm {adjacent} }}$

${\displaystyle \csc \theta ={\frac {1}{\sin \theta }}={\mathrm {hypoteneuse} \over \mathrm {opposite} }}$

Table of Common Angles

 Degrees Radians Sine Cosine Tangent ${\displaystyle 0\,}$ ${\displaystyle 0\,}$ ${\displaystyle {0}\,}$ ${\displaystyle {1}\,}$ ${\displaystyle 0\,}$ ${\displaystyle 30\,}$ ${\displaystyle {\pi \over 6}}$ ${\displaystyle {1 \over 2}}$ ${\displaystyle {{\sqrt {3}} \over 2}}$ ${\displaystyle {1 \over {\sqrt {3}}}}$ ${\displaystyle 45\,}$ ${\displaystyle {\pi \over 4}}$ ${\displaystyle {{\sqrt {2}} \over 2}}$ ${\displaystyle {{\sqrt {2}} \over 2}}$ ${\displaystyle 1\,}$ ${\displaystyle 60\,}$ ${\displaystyle {\pi \over 3}}$ ${\displaystyle {{\sqrt {3}} \over 2}}$ ${\displaystyle {1 \over 2}}$ ${\displaystyle {\sqrt {3}}}$ ${\displaystyle 90\,}$ ${\displaystyle {\pi \over 2}}$ ${\displaystyle 1\,}$ ${\displaystyle 0\,}$ ${\displaystyle undefined\,}$

Identities

${\displaystyle \tan \theta ={\sin \theta \over \cos \theta }}$

${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1\,}$

${\displaystyle 1+\tan ^{2}\theta ={1 \over \cos ^{2}\theta }}$

${\displaystyle \tan 2\theta ={2\tan \theta \over {1-\tan ^{2}\theta }}}$