9514 1404 393
Answer:
0.927184
Step-by-step explanation:
Assuming you want cos(338°), it is equal to cos(22°). This value is an irrational root of a 26th-degree polynomial. (Neither the polynomial, nor the root are easy to find by hand.)
The values of trig functions for 30° and 45° are well-known, as are half-angle and angle sum/difference formulas. The nearest angle for which cosine is relatively easy to find is 22.5°. A 2nd-order Maclaurin series expansion of the cosine function about that angle gives the approximation ...
cos(x) ≈ cos(π/8)(1 -π²/128) +π/8·sin(π/8) +(π/8·cos(π/8) -sin(π/8))x -1/2·cos(π/8)x²
For your angle of 22°, the value of x needed in this expression is 11π/90. The values of cos(π/8) and sin(π/8) are the irrational numbers ...
cos(π/8) = √(2 +√2)/2 ≈ 0.92387953
sin(π/8) = √(2 -√2)/2 ≈ 0.38268343
If you use the approximation pi ≈ 355/113, both for the above formula and for the angle of interest, this reduces to the sum ...
cos(22°) ≈ 71√(2-√2)/16272 +132383951√(2+√2)/264777984
This would be tedious to calculate by hand, but is not impossible. The value this sum produces will be correct when rounded to 7 decimal digits.
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Before calculators, trig function values were found in published tables. These days, tables are available on-line. Previously, they were published in books, usually along with information that helped with interpolation between table values, when necessary. The attachment shows a portion of an online cosine table suitable for answering your question.
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Additional comment
Before calculators, mathematicians spent lifetimes calculating accurate trig tables. For all but a very few angles, the cosine function of rational degree values will be an irrational number, as it is for cos(22°).
