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Can we solve for $delta$ in $fracÀ2 - fracRrβ = arcsinbiggl(frac(R+r)sin(δ)rbiggr)+fracRrδ$?
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Solve for $delta$:
$$fracpi2 - fracRrβ = arcsinleft(frac(R+r)sin deltarright)+fracRrδ$$
My problem is that I can't comprehend how to get both $delta$s out of a trig operation at the same time. If I apply 'sine' to both sides, I understand the 'arcsine' will be canceled, but does this operation apply 'sine' to $fracRrdelta$ on the right, making it $sinleft(fracRrdeltaright)$ ???
Is this problem a question of a trigonometric derivative?
My second question is perhaps a more important one. I do not have a textbook pertaining to trigonometry, much less trigonometric derivatives. The math book I currently own is about matrices and calculus, but only having to do with 'regular' derivatives; no trigonometry derivatives, obviously.
What is a good textbook for trigonometry (regular trig-algebra)?
What is a good textbook for trigonometry derivatives? Is there a textbook that contains both of these subjects?
Thanks is advance.
algebra-precalculus trigonometry
edited yesterday
Blue
43.6k868141
asked yesterday
user101434
44
add a comment |Â
up vote
0
down vote
favorite
Solve for $delta$:
$$fracpi2 - fracRrβ = arcsinleft(frac(R+r)sin deltarright)+fracRrδ$$
My problem is that I can't comprehend how to get both $delta$s out of a trig operation at the same time. If I apply 'sine' to both sides, I understand the 'arcsine' will be canceled, but does this operation apply 'sine' to $fracRrdelta$ on the right, making it $sinleft(fracRrdeltaright)$ ???
Is this problem a question of a trigonometric derivative?
My second question is perhaps a more important one. I do not have a textbook pertaining to trigonometry, much less trigonometric derivatives. The math book I currently own is about matrices and calculus, but only having to do with 'regular' derivatives; no trigonometry derivatives, obviously.
What is a good textbook for trigonometry (regular trig-algebra)?
What is a good textbook for trigonometry derivatives? Is there a textbook that contains both of these subjects?
Thanks is advance.
algebra-precalculus trigonometry
edited yesterday
Blue
43.6k868141
asked yesterday
user101434
44
You need a numerical method. Remember that $x=cos(x)$ does not show analytical solutions. If you give test values for $r,R,beta$, I could shos you.
– Claude Leibovici
yesterday
You should ask for textbook recommendations in a separate question. That said, do a search first. I'm pretty sure that people have asked for such recommendations a few times.
– Blue
yesterday
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Solve for $delta$:
$$fracpi2 - fracRrβ = arcsinleft(frac(R+r)sin deltarright)+fracRrδ$$
My problem is that I can't comprehend how to get both $delta$s out of a trig operation at the same time. If I apply 'sine' to both sides, I understand the 'arcsine' will be canceled, but does this operation apply 'sine' to $fracRrdelta$ on the right, making it $sinleft(fracRrdeltaright)$ ???
Is this problem a question of a trigonometric derivative?
My second question is perhaps a more important one. I do not have a textbook pertaining to trigonometry, much less trigonometric derivatives. The math book I currently own is about matrices and calculus, but only having to do with 'regular' derivatives; no trigonometry derivatives, obviously.
What is a good textbook for trigonometry (regular trig-algebra)?
What is a good textbook for trigonometry derivatives? Is there a textbook that contains both of these subjects?
Thanks is advance.
algebra-precalculus trigonometry
edited yesterday
Blue
43.6k868141
asked yesterday
user101434
44
Solve for $delta$:
$$fracpi2 - fracRrβ = arcsinleft(frac(R+r)sin deltarright)+fracRrδ$$
My problem is that I can't comprehend how to get both $delta$s out of a trig operation at the same time. If I apply 'sine' to both sides, I understand the 'arcsine' will be canceled, but does this operation apply 'sine' to $fracRrdelta$ on the right, making it $sinleft(fracRrdeltaright)$ ???
Is this problem a question of a trigonometric derivative?
My second question is perhaps a more important one. I do not have a textbook pertaining to trigonometry, much less trigonometric derivatives. The math book I currently own is about matrices and calculus, but only having to do with 'regular' derivatives; no trigonometry derivatives, obviously.
What is a good textbook for trigonometry (regular trig-algebra)?
What is a good textbook for trigonometry derivatives? Is there a textbook that contains both of these subjects?
Thanks is advance.
algebra-precalculus trigonometry
edited yesterday
Blue
43.6k868141
asked yesterday
user101434
44
edited yesterday
Blue
43.6k868141
edited yesterday
Blue
43.6k868141
edited yesterday
Blue
43.6k868141
43.6k868141
asked yesterday
user101434
44
asked yesterday
user101434
44
asked yesterday
user101434
44
44
You need a numerical method. Remember that $x=cos(x)$ does not show analytical solutions. If you give test values for $r,R,beta$, I could shos you.
– Claude Leibovici
yesterday
You should ask for textbook recommendations in a separate question. That said, do a search first. I'm pretty sure that people have asked for such recommendations a few times.
– Blue
yesterday
add a comment |Â
You need a numerical method. Remember that $x=cos(x)$ does not show analytical solutions. If you give test values for $r,R,beta$, I could shos you.
– Claude Leibovici
yesterday
You should ask for textbook recommendations in a separate question. That said, do a search first. I'm pretty sure that people have asked for such recommendations a few times.
– Blue
yesterday
You need a numerical method. Remember that $x=cos(x)$ does not show analytical solutions. If you give test values for $r,R,beta$, I could shos you.
– Claude Leibovici
yesterday
You need a numerical method. Remember that $x=cos(x)$ does not show analytical solutions. If you give test values for $r,R,beta$, I could shos you.
– Claude Leibovici
yesterday
You should ask for textbook recommendations in a separate question. That said, do a search first. I'm pretty sure that people have asked for such recommendations a few times.
– Blue
yesterday
You should ask for textbook recommendations in a separate question. That said, do a search first. I'm pretty sure that people have asked for such recommendations a few times.
– Blue
yesterday
add a comment |Â
1 Answer
1
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Letting $Q = R/r$ and rearranging a bit gives
$$
fracpi2 - Qbeta - Qdelta = arcsin bigl( (Q + 1) sin delta bigr)
$$
then taking $sin(cdot)$ of both sides and using $sin (pi/2 - x) = cos x$ results in
$$
cos bigl( Q(beta + delta) bigr) = (Q + 1) sin delta.
$$
If $Q = 1$ then the equation can be solved exactly. Otherwise, you can rewrite the equation as
$$
delta = f(delta) = arcsin left( fraccos bigl( Q(beta + delta) bigr)Q + 1 right)
$$
and repeatedly apply $f$ to some initial guess for a solution $delta$. It turns out that this fixed-point iteration converges in this particular case.
answered yesterday
D. G.
464
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Letting $Q = R/r$ and rearranging a bit gives
$$
fracpi2 - Qbeta - Qdelta = arcsin bigl( (Q + 1) sin delta bigr)
$$
then taking $sin(cdot)$ of both sides and using $sin (pi/2 - x) = cos x$ results in
$$
cos bigl( Q(beta + delta) bigr) = (Q + 1) sin delta.
$$
If $Q = 1$ then the equation can be solved exactly. Otherwise, you can rewrite the equation as
$$
delta = f(delta) = arcsin left( fraccos bigl( Q(beta + delta) bigr)Q + 1 right)
$$
and repeatedly apply $f$ to some initial guess for a solution $delta$. It turns out that this fixed-point iteration converges in this particular case.
answered yesterday
D. G.
464
add a comment |Â
up vote
2
down vote
Letting $Q = R/r$ and rearranging a bit gives
$$
fracpi2 - Qbeta - Qdelta = arcsin bigl( (Q + 1) sin delta bigr)
$$
then taking $sin(cdot)$ of both sides and using $sin (pi/2 - x) = cos x$ results in
$$
cos bigl( Q(beta + delta) bigr) = (Q + 1) sin delta.
$$
If $Q = 1$ then the equation can be solved exactly. Otherwise, you can rewrite the equation as
$$
delta = f(delta) = arcsin left( fraccos bigl( Q(beta + delta) bigr)Q + 1 right)
$$
and repeatedly apply $f$ to some initial guess for a solution $delta$. It turns out that this fixed-point iteration converges in this particular case.
answered yesterday
D. G.
464
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Letting $Q = R/r$ and rearranging a bit gives
$$
fracpi2 - Qbeta - Qdelta = arcsin bigl( (Q + 1) sin delta bigr)
$$
then taking $sin(cdot)$ of both sides and using $sin (pi/2 - x) = cos x$ results in
$$
cos bigl( Q(beta + delta) bigr) = (Q + 1) sin delta.
$$
If $Q = 1$ then the equation can be solved exactly. Otherwise, you can rewrite the equation as
$$
delta = f(delta) = arcsin left( fraccos bigl( Q(beta + delta) bigr)Q + 1 right)
$$
and repeatedly apply $f$ to some initial guess for a solution $delta$. It turns out that this fixed-point iteration converges in this particular case.
answered yesterday
D. G.
464
Letting $Q = R/r$ and rearranging a bit gives
$$
fracpi2 - Qbeta - Qdelta = arcsin bigl( (Q + 1) sin delta bigr)
$$
then taking $sin(cdot)$ of both sides and using $sin (pi/2 - x) = cos x$ results in
$$
cos bigl( Q(beta + delta) bigr) = (Q + 1) sin delta.
$$
If $Q = 1$ then the equation can be solved exactly. Otherwise, you can rewrite the equation as
$$
delta = f(delta) = arcsin left( fraccos bigl( Q(beta + delta) bigr)Q + 1 right)
$$
and repeatedly apply $f$ to some initial guess for a solution $delta$. It turns out that this fixed-point iteration converges in this particular case.
answered yesterday
D. G.
464
answered yesterday
D. G.
464
answered yesterday
D. G.
464
answered yesterday
D. G.
464
464
add a comment |Â
add a comment |Â
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You need a numerical method. Remember that $x=cos(x)$ does not show analytical solutions. If you give test values for $r,R,beta$, I could shos you.
– Claude Leibovici
yesterday
You should ask for textbook recommendations in a separate question. That said, do a search first. I'm pretty sure that people have asked for such recommendations a few times.
– Blue
yesterday