(that is, without computation or tables).
Hint 1. Find closed form expressions for
See Sublime Triangle
Cos 36 degrees
Closed Form for Some Values of Cosine and Sine
Evaluation once you have the closed form expressions
The hints below may be useful for finding closed form expressions OR for approaching the problem algebraically.
Hint 2. Consider the isosceles triangle with angles 36, 72, 72.
See Sublime Triangle.
Hint 3. Consider trigonometric identities for half angles and double angles.
Hint 4. Consider the equation