Mixing
How to solve equations like $alpha sin x -betasin 2x +gamma=0 $
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Can I solve this equation without Newton-Raphson method?
I have $alpha=47.02$ $beta=112.5$ and $gamma=50$.
When I have to use Newton-Rapson to solve trigonometric equations ?
I will greatly appreciate your answers.
calculus trigonometry approximation newton-raphson
asked Jul 19 at 21:33
Mauricio J. S.
61
add a comment |Â
up vote
1
down vote
favorite
Can I solve this equation without Newton-Raphson method?
I have $alpha=47.02$ $beta=112.5$ and $gamma=50$.
When I have to use Newton-Rapson to solve trigonometric equations ?
I will greatly appreciate your answers.
calculus trigonometry approximation newton-raphson
asked Jul 19 at 21:33
Mauricio J. S.
61
1
Are you asking for a type of closed form for x? Are you asking for different numerical methods?
– Mason
Jul 19 at 21:36
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Can I solve this equation without Newton-Raphson method?
I have $alpha=47.02$ $beta=112.5$ and $gamma=50$.
When I have to use Newton-Rapson to solve trigonometric equations ?
I will greatly appreciate your answers.
calculus trigonometry approximation newton-raphson
asked Jul 19 at 21:33
Mauricio J. S.
61
Can I solve this equation without Newton-Raphson method?
I have $alpha=47.02$ $beta=112.5$ and $gamma=50$.
When I have to use Newton-Rapson to solve trigonometric equations ?
I will greatly appreciate your answers.
calculus trigonometry approximation newton-raphson
asked Jul 19 at 21:33
Mauricio J. S.
61
asked Jul 19 at 21:33
Mauricio J. S.
61
asked Jul 19 at 21:33
Mauricio J. S.
61
asked Jul 19 at 21:33
Mauricio J. S.
61
61
1
Are you asking for a type of closed form for x? Are you asking for different numerical methods?
– Mason
Jul 19 at 21:36
add a comment |Â
1
Are you asking for a type of closed form for x? Are you asking for different numerical methods?
– Mason
Jul 19 at 21:36
1
1
Are you asking for a type of closed form for x? Are you asking for different numerical methods?
– Mason
Jul 19 at 21:36
Are you asking for a type of closed form for x? Are you asking for different numerical methods?
– Mason
Jul 19 at 21:36
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
3
down vote
Hint:
Substitute
$$sin(x)=frac2t1+t^2$$
$$cos(x)=frac1-t^21+t^2$$
the so called Weierstrass substitution.
answered Jul 19 at 21:46
Dr. Sonnhard Graubner
66.8k32659
2
This is erroneously called the Weierstrass substitution. James Stewart's widely used calculus textbooks say that Weierstrass pointed it out. Stewart claimed to have learned that elsewhere but said he couldn't remember where. That's the blind leading the blind. I'd call it the tangent half-angle substitution, since $t = tandfrac x 2. qquad$
– Michael Hardy
Jul 19 at 22:06
@MichaelHardy, Indeed! I would say that your comment might only be improved by a link to wiki en.wikipedia.org/wiki/Tangent_half-angle_substitution which confirms this.
– Mason
Jul 19 at 22:09
Ok, let's say tan half angle substitution.
– Dr. Sonnhard Graubner
Jul 19 at 22:25
add a comment |Â
up vote
1
down vote
Alt. hint: Â write it as $, 2 beta sin x cos x = alpha sin x + gamma,$, and square both sides. With $,s = sin x,$ the equation then becomes a depressed quartic: $;4 beta^2 s^2(1-s^2) = (alpha s + gamma)^2,$.
answered Jul 20 at 0:15
dxiv
54.2k64797
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Hint:
Substitute
$$sin(x)=frac2t1+t^2$$
$$cos(x)=frac1-t^21+t^2$$
the so called Weierstrass substitution.
answered Jul 19 at 21:46
Dr. Sonnhard Graubner
66.8k32659
2
This is erroneously called the Weierstrass substitution. James Stewart's widely used calculus textbooks say that Weierstrass pointed it out. Stewart claimed to have learned that elsewhere but said he couldn't remember where. That's the blind leading the blind. I'd call it the tangent half-angle substitution, since $t = tandfrac x 2. qquad$
– Michael Hardy
Jul 19 at 22:06
@MichaelHardy, Indeed! I would say that your comment might only be improved by a link to wiki en.wikipedia.org/wiki/Tangent_half-angle_substitution which confirms this.
– Mason
Jul 19 at 22:09
Ok, let's say tan half angle substitution.
– Dr. Sonnhard Graubner
Jul 19 at 22:25
add a comment |Â
up vote
3
down vote
Hint:
Substitute
$$sin(x)=frac2t1+t^2$$
$$cos(x)=frac1-t^21+t^2$$
the so called Weierstrass substitution.
answered Jul 19 at 21:46
Dr. Sonnhard Graubner
66.8k32659
2
This is erroneously called the Weierstrass substitution. James Stewart's widely used calculus textbooks say that Weierstrass pointed it out. Stewart claimed to have learned that elsewhere but said he couldn't remember where. That's the blind leading the blind. I'd call it the tangent half-angle substitution, since $t = tandfrac x 2. qquad$
– Michael Hardy
Jul 19 at 22:06
@MichaelHardy, Indeed! I would say that your comment might only be improved by a link to wiki en.wikipedia.org/wiki/Tangent_half-angle_substitution which confirms this.
– Mason
Jul 19 at 22:09
Ok, let's say tan half angle substitution.
– Dr. Sonnhard Graubner
Jul 19 at 22:25
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Hint:
Substitute
$$sin(x)=frac2t1+t^2$$
$$cos(x)=frac1-t^21+t^2$$
the so called Weierstrass substitution.
answered Jul 19 at 21:46
Dr. Sonnhard Graubner
66.8k32659
Hint:
Substitute
$$sin(x)=frac2t1+t^2$$
$$cos(x)=frac1-t^21+t^2$$
the so called Weierstrass substitution.
answered Jul 19 at 21:46
Dr. Sonnhard Graubner
66.8k32659
edited Jul 19 at 21:58
answered Jul 19 at 21:46
Dr. Sonnhard Graubner
66.8k32659
answered Jul 19 at 21:46
Dr. Sonnhard Graubner
66.8k32659
answered Jul 19 at 21:46
Dr. Sonnhard Graubner
66.8k32659
66.8k32659
2
This is erroneously called the Weierstrass substitution. James Stewart's widely used calculus textbooks say that Weierstrass pointed it out. Stewart claimed to have learned that elsewhere but said he couldn't remember where. That's the blind leading the blind. I'd call it the tangent half-angle substitution, since $t = tandfrac x 2. qquad$
– Michael Hardy
Jul 19 at 22:06
@MichaelHardy, Indeed! I would say that your comment might only be improved by a link to wiki en.wikipedia.org/wiki/Tangent_half-angle_substitution which confirms this.
– Mason
Jul 19 at 22:09
Ok, let's say tan half angle substitution.
– Dr. Sonnhard Graubner
Jul 19 at 22:25
add a comment |Â
2
This is erroneously called the Weierstrass substitution. James Stewart's widely used calculus textbooks say that Weierstrass pointed it out. Stewart claimed to have learned that elsewhere but said he couldn't remember where. That's the blind leading the blind. I'd call it the tangent half-angle substitution, since $t = tandfrac x 2. qquad$
– Michael Hardy
Jul 19 at 22:06
@MichaelHardy, Indeed! I would say that your comment might only be improved by a link to wiki en.wikipedia.org/wiki/Tangent_half-angle_substitution which confirms this.
– Mason
Jul 19 at 22:09
Ok, let's say tan half angle substitution.
– Dr. Sonnhard Graubner
Jul 19 at 22:25
2
2
This is erroneously called the Weierstrass substitution. James Stewart's widely used calculus textbooks say that Weierstrass pointed it out. Stewart claimed to have learned that elsewhere but said he couldn't remember where. That's the blind leading the blind. I'd call it the tangent half-angle substitution, since $t = tandfrac x 2. qquad$
– Michael Hardy
Jul 19 at 22:06
This is erroneously called the Weierstrass substitution. James Stewart's widely used calculus textbooks say that Weierstrass pointed it out. Stewart claimed to have learned that elsewhere but said he couldn't remember where. That's the blind leading the blind. I'd call it the tangent half-angle substitution, since $t = tandfrac x 2. qquad$
– Michael Hardy
Jul 19 at 22:06
@MichaelHardy, Indeed! I would say that your comment might only be improved by a link to wiki en.wikipedia.org/wiki/Tangent_half-angle_substitution which confirms this.
– Mason
Jul 19 at 22:09
@MichaelHardy, Indeed! I would say that your comment might only be improved by a link to wiki en.wikipedia.org/wiki/Tangent_half-angle_substitution which confirms this.
– Mason
Jul 19 at 22:09
Ok, let's say tan half angle substitution.
– Dr. Sonnhard Graubner
Jul 19 at 22:25
Ok, let's say tan half angle substitution.
– Dr. Sonnhard Graubner
Jul 19 at 22:25
add a comment |Â
up vote
1
down vote
Alt. hint: Â write it as $, 2 beta sin x cos x = alpha sin x + gamma,$, and square both sides. With $,s = sin x,$ the equation then becomes a depressed quartic: $;4 beta^2 s^2(1-s^2) = (alpha s + gamma)^2,$.
answered Jul 20 at 0:15
dxiv
54.2k64797
add a comment |Â
up vote
1
down vote
Alt. hint: Â write it as $, 2 beta sin x cos x = alpha sin x + gamma,$, and square both sides. With $,s = sin x,$ the equation then becomes a depressed quartic: $;4 beta^2 s^2(1-s^2) = (alpha s + gamma)^2,$.
answered Jul 20 at 0:15
dxiv
54.2k64797
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Alt. hint: Â write it as $, 2 beta sin x cos x = alpha sin x + gamma,$, and square both sides. With $,s = sin x,$ the equation then becomes a depressed quartic: $;4 beta^2 s^2(1-s^2) = (alpha s + gamma)^2,$.
answered Jul 20 at 0:15
dxiv
54.2k64797
Alt. hint: Â write it as $, 2 beta sin x cos x = alpha sin x + gamma,$, and square both sides. With $,s = sin x,$ the equation then becomes a depressed quartic: $;4 beta^2 s^2(1-s^2) = (alpha s + gamma)^2,$.
answered Jul 20 at 0:15
dxiv
54.2k64797
answered Jul 20 at 0:15
dxiv
54.2k64797
answered Jul 20 at 0:15
dxiv
54.2k64797
answered Jul 20 at 0:15
dxiv
54.2k64797
54.2k64797
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2857066%2fhow-to-solve-equations-like-alpha-sin-x-beta-sin-2x-gamma-0%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
Are you asking for a type of closed form for x? Are you asking for different numerical methods?
– Mason
Jul 19 at 21:36