for example, I would like to solve for sin 17.5°, tan17.5°, cos 17.5°, and the other trig values. What methods should I use whenever I encounter these kinds of questions? Are there numerous possible ways? I am very interested.
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1All values of sine and cosine are on the unit circle. Perhaps you mean that these particular angles are not drawn in the typical diagram showing values of sine and cosine for $30^\circ,$ $45^\circ,$ $60^\circ,$ etc. There are other angles whose trig functions can be represented exactly in finitely many operations with square roots and arithmetic operations, but $17.5^\circ$ is not such an angle. – David K Dec 01 '22 at 03:28
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There is some relevant information in the answers to https://math.stackexchange.com/q/395600/139123 -- It is about calculators and computers, but there are no simpler methods that you can use by hand. The way we used to solve these functions by hand before calculators was to look the answer up in a large table of numbers. – David K Dec 01 '22 at 03:38
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thank you verymuch! – salierii Dec 01 '22 at 03:53
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If I am correctly interpreting your concern, you are asking for a simple rule to calculate $\sin(x)$, $\cos(x)$, $\tan(x)$ (and so forth), for "random" values of $x$.
IMHO, your starting point would be the well-known Left Hand Trick so that you can calculate "by hands" every value of the above mentioned trigonometric functions which is a multiple of $\frac{\pi}{6}=30°$. Thus, you can easily evaluate an upper and a lower bound arising from $x:=0°$ and $x:=30°$ (e.g., you could deduce that $\sin(17.5°)>\sin(15°)>\frac{\sin(30°)}{2}=0.25$ and $\sin(17.5°)<\sin(30°)=0.5$).
If this is not enough, just use an online calculator that instantly returns the approximated value and also the exact result (if available), as $\sin(17.5°)$ or $\sin(15°)$ (see "exact result").
Marco Ripà
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