.Multiples of 15 Degrees
Trigonometry texts always include material early in the course on finding the
exact values of trig functions of the angles 0°, 30°, 45°, 60°, and 90°.
It is also true that by a similar argument, exact values of trig functions of
the angles 15° and 75° may also be found. These angles are
equivalent to unit circle arc lengths that are multiples of
.
Trig Values of Multiples of
:
| θ | sin θ | cos θ | tan θ | cot θ | sec θ | csc θ |
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Proof: Using the angle difference identity for sine, we have:
The derivation of the other values is similar.♦
If a half-angle formula is used, then the result 
would be obtained. But these two results are identical, since:
Alternative Proof: Let triangle TPU be inscribed in rectangle PQRS, so that angle PTU is 90 degrees, QPT is 30 degrees, angle TPU is 45 degrees, and segment PU has length 1. Then angle UPS is 15 degrees, and angle RTU is 30 degrees.
By the Triangle Ratios Theorem, we have:
The derivation of the other values is similar.♦