Mixing
Roots of equation involving both polynomials and trigonometric functions
$begingroup$
I have to determine for which values of $x$ that the velocity vector is orthogonal to the acceleration vector, the position is given by:
$(3 cos(t), - sin(3t), 2t^3 - t^2)$,
I then use that $u cdot v = 0$ if $u$ and $v$ is orthogonal. The dot product of the acceleration and velocity gives the following equation (if I have done the differentiation correctly):
$9sinleft(tright)cosleft(tright)-27cosleft(3tright)sinleft(3tright)+4tleft(18t^2-9t+1right) = 0$.
$t = 0$ (or position $x = (3,0,0)$) is clearly a root but i can't find the other two. Is there a general way of computing roots of this sort of function analytically?
real-analysis trigonometry roots
edited Jan 21 at 14:52
Andreas
8,2431137
asked Jan 21 at 14:06
LoggomanLoggoman
63
$endgroup$
|
show 3 more comments
$begingroup$
I have to determine for which values of $x$ that the velocity vector is orthogonal to the acceleration vector, the position is given by:
$(3 cos(t), - sin(3t), 2t^3 - t^2)$,
I then use that $u cdot v = 0$ if $u$ and $v$ is orthogonal. The dot product of the acceleration and velocity gives the following equation (if I have done the differentiation correctly):
$9sinleft(tright)cosleft(tright)-27cosleft(3tright)sinleft(3tright)+4tleft(18t^2-9t+1right) = 0$.
$t = 0$ (or position $x = (3,0,0)$) is clearly a root but i can't find the other two. Is there a general way of computing roots of this sort of function analytically?
real-analysis trigonometry roots
edited Jan 21 at 14:52
Andreas
8,2431137
asked Jan 21 at 14:06
LoggomanLoggoman
63
$endgroup$
1
$begingroup$
Except in very special cases, equations involving both trig functions and polynomials cannot be solved explicitly in the sense you probably intend. Are you SURE this is not a calculator exercise, where you solve by the use of a graphing calculator?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:15
$begingroup$
@DaveL.Renfro I had a small typo in the equation, it is corrected now. But yes, this was given to students as a question in multivariate calculus on a test.
$endgroup$
– Loggoman
Jan 21 at 14:19
1
$begingroup$
@Blue You are correct, thanks.
$endgroup$
– Loggoman
Jan 21 at 14:33
$begingroup$
But yes, this was given to students as a question in multivariate calculus on a test. --- Were calculators allowed to be used on the test?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:46
$begingroup$
@DaveL.Renfro No, only pencil and eraser.
$endgroup$
– Loggoman
Jan 21 at 14:50
|
show 3 more comments
$begingroup$
I have to determine for which values of $x$ that the velocity vector is orthogonal to the acceleration vector, the position is given by:
$(3 cos(t), - sin(3t), 2t^3 - t^2)$,
I then use that $u cdot v = 0$ if $u$ and $v$ is orthogonal. The dot product of the acceleration and velocity gives the following equation (if I have done the differentiation correctly):
$9sinleft(tright)cosleft(tright)-27cosleft(3tright)sinleft(3tright)+4tleft(18t^2-9t+1right) = 0$.
$t = 0$ (or position $x = (3,0,0)$) is clearly a root but i can't find the other two. Is there a general way of computing roots of this sort of function analytically?
real-analysis trigonometry roots
edited Jan 21 at 14:52
Andreas
8,2431137
asked Jan 21 at 14:06
LoggomanLoggoman
63
$endgroup$
I have to determine for which values of $x$ that the velocity vector is orthogonal to the acceleration vector, the position is given by:
$(3 cos(t), - sin(3t), 2t^3 - t^2)$,
I then use that $u cdot v = 0$ if $u$ and $v$ is orthogonal. The dot product of the acceleration and velocity gives the following equation (if I have done the differentiation correctly):
$9sinleft(tright)cosleft(tright)-27cosleft(3tright)sinleft(3tright)+4tleft(18t^2-9t+1right) = 0$.
$t = 0$ (or position $x = (3,0,0)$) is clearly a root but i can't find the other two. Is there a general way of computing roots of this sort of function analytically?
real-analysis trigonometry roots
real-analysis trigonometry roots
edited Jan 21 at 14:52
Andreas
8,2431137
asked Jan 21 at 14:06
LoggomanLoggoman
63
edited Jan 21 at 14:52
Andreas
8,2431137
asked Jan 21 at 14:06
LoggomanLoggoman
63
edited Jan 21 at 14:52
Andreas
8,2431137
edited Jan 21 at 14:52
Andreas
8,2431137
edited Jan 21 at 14:52
Andreas
8,2431137
8,2431137
asked Jan 21 at 14:06
LoggomanLoggoman
63
asked Jan 21 at 14:06
LoggomanLoggoman
63
asked Jan 21 at 14:06
LoggomanLoggoman
63
63
1
$begingroup$
Except in very special cases, equations involving both trig functions and polynomials cannot be solved explicitly in the sense you probably intend. Are you SURE this is not a calculator exercise, where you solve by the use of a graphing calculator?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:15
$begingroup$
@DaveL.Renfro I had a small typo in the equation, it is corrected now. But yes, this was given to students as a question in multivariate calculus on a test.
$endgroup$
– Loggoman
Jan 21 at 14:19
1
$begingroup$
@Blue You are correct, thanks.
$endgroup$
– Loggoman
Jan 21 at 14:33
$begingroup$
But yes, this was given to students as a question in multivariate calculus on a test. --- Were calculators allowed to be used on the test?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:46
$begingroup$
@DaveL.Renfro No, only pencil and eraser.
$endgroup$
– Loggoman
Jan 21 at 14:50
|
show 3 more comments
1
$begingroup$
Except in very special cases, equations involving both trig functions and polynomials cannot be solved explicitly in the sense you probably intend. Are you SURE this is not a calculator exercise, where you solve by the use of a graphing calculator?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:15
$begingroup$
@DaveL.Renfro I had a small typo in the equation, it is corrected now. But yes, this was given to students as a question in multivariate calculus on a test.
$endgroup$
– Loggoman
Jan 21 at 14:19
1
$begingroup$
@Blue You are correct, thanks.
$endgroup$
– Loggoman
Jan 21 at 14:33
$begingroup$
But yes, this was given to students as a question in multivariate calculus on a test. --- Were calculators allowed to be used on the test?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:46
$begingroup$
@DaveL.Renfro No, only pencil and eraser.
$endgroup$
– Loggoman
Jan 21 at 14:50
1
1
$begingroup$
Except in very special cases, equations involving both trig functions and polynomials cannot be solved explicitly in the sense you probably intend. Are you SURE this is not a calculator exercise, where you solve by the use of a graphing calculator?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:15
$begingroup$
Except in very special cases, equations involving both trig functions and polynomials cannot be solved explicitly in the sense you probably intend. Are you SURE this is not a calculator exercise, where you solve by the use of a graphing calculator?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:15
$begingroup$
@DaveL.Renfro I had a small typo in the equation, it is corrected now. But yes, this was given to students as a question in multivariate calculus on a test.
$endgroup$
– Loggoman
Jan 21 at 14:19
$begingroup$
@DaveL.Renfro I had a small typo in the equation, it is corrected now. But yes, this was given to students as a question in multivariate calculus on a test.
$endgroup$
– Loggoman
Jan 21 at 14:19
1
1
$begingroup$
@Blue You are correct, thanks.
$endgroup$
– Loggoman
Jan 21 at 14:33
$begingroup$
@Blue You are correct, thanks.
$endgroup$
– Loggoman
Jan 21 at 14:33
$begingroup$
But yes, this was given to students as a question in multivariate calculus on a test. --- Were calculators allowed to be used on the test?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:46
$begingroup$
But yes, this was given to students as a question in multivariate calculus on a test. --- Were calculators allowed to be used on the test?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:46
$begingroup$
@DaveL.Renfro No, only pencil and eraser.
$endgroup$
– Loggoman
Jan 21 at 14:50
$begingroup$
@DaveL.Renfro No, only pencil and eraser.
$endgroup$
– Loggoman
Jan 21 at 14:50
|
show 3 more comments
0
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$begingroup$
Except in very special cases, equations involving both trig functions and polynomials cannot be solved explicitly in the sense you probably intend. Are you SURE this is not a calculator exercise, where you solve by the use of a graphing calculator?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:15
$begingroup$
@DaveL.Renfro I had a small typo in the equation, it is corrected now. But yes, this was given to students as a question in multivariate calculus on a test.
$endgroup$
– Loggoman
Jan 21 at 14:19
1
$begingroup$
@Blue You are correct, thanks.
$endgroup$
– Loggoman
Jan 21 at 14:33
$begingroup$
But yes, this was given to students as a question in multivariate calculus on a test. --- Were calculators allowed to be used on the test?
$endgroup$
– Dave L. Renfro
Jan 21 at 14:46
$begingroup$
@DaveL.Renfro No, only pencil and eraser.
$endgroup$
– Loggoman
Jan 21 at 14:50