Trigonometry/The sine of 18 degrees

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The trigonometric functions of angles that are multiples of 3º can all be expressed as surds. This page will only deal with angles that are multiples of 18º.

Consider the equation

.

Clearly, the only solutions are that 5x = 90º, 90º + 360º, 90º + 2x360º, 90º + 3x360º ... Thus x = 18º, 90º, 162º, 234º or 306º. (There is no need to go further, since increasing x by 360º does not affect its sine.) is, after using various simplifications, sin(18º) twice, 1, -sin(54º) twice.

This equation can be solved in another way. Using the formula for , we can expand to get

If we write , this becomes a quintic equation in s,

i.e.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle (s-1)(4s^2+2s-1)^2=0}

so either (corresponding to x = 90º) or

(corresponding to x = 18º or 54º). Thus

.
.


Exercise: More angles

By considering the equation

show that

.
.

The cosines are found immediately, e.g. cos(18º) = sin(72º).