trigonometric equation: solve sin 3x + tan x = 3
- Thread starter sickplaya
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In this case, a rough estimate can be obtained by a little thinking on the range of the two pieces.
sin(3x) is restricted to [-1,1]
If that left-hand side is going to get anywhere near 3, the tangent piece will have to be rather hefty in the positive direction. Where does the tangent get big in the positive direction? Somewhere around \(\displaystyle \frac{\pi}{2}\,and\,\frac{3\,\pi}{2}\)? The solutions should be a little less than those values.
You can play with identities until the cows come home. Something may fall out. To me, it appears to result in a 8th degree polynomial on cos(x). That can't be pretty.
sin(3x) is restricted to [-1,1]
If that left-hand side is going to get anywhere near 3, the tangent piece will have to be rather hefty in the positive direction. Where does the tangent get big in the positive direction? Somewhere around \(\displaystyle \frac{\pi}{2}\,and\,\frac{3\,\pi}{2}\)? The solutions should be a little less than those values.
You can play with identities until the cows come home. Something may fall out. To me, it appears to result in a 8th degree polynomial on cos(x). That can't be pretty.
Are you familiar with Newton's method?. I tried it with an initial estimate of 1.5 and arrived at x=1.30703159871 as one solution. Try other intial estimates and you'll probably get other solutions. I believe 4.34016557987 is another.