Mixing
limit involving trig
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
I am trying to figure out what is wrong with my solution to the following, posted below 
Suppose, without loss of generality that the circle has radius $1$. Then $theta = s$. From the law of cosines, we have $d=2+2cos(theta)$. From this we get that
$$lim_thetarightarrow 0fracsd=lim_thetarightarrow 0fractheta2-2cos(theta)=lim_thetarightarrow 0frac12sin(theta)=infty.
$$
However, I am told that the answer is $1$. What is wrong with my reasoning?
calculus
edited Jul 31 at 18:18
gt6989b
30.2k22148
asked Jul 31 at 18:14
ponchan
34719
add a comment |Â
up vote
0
down vote
favorite
I am trying to figure out what is wrong with my solution to the following, posted below
Suppose, without loss of generality that the circle has radius $1$. Then $theta = s$. From the law of cosines, we have $d=2+2cos(theta)$. From this we get that
$$lim_thetarightarrow 0fracsd=lim_thetarightarrow 0fractheta2-2cos(theta)=lim_thetarightarrow 0frac12sin(theta)=infty.
$$
However, I am told that the answer is $1$. What is wrong with my reasoning?
calculus
edited Jul 31 at 18:18
gt6989b
30.2k22148
asked Jul 31 at 18:14
ponchan
34719
3
$d=2+2costheta$ is certainly wrong, since that predicts $d=4$ when $theta = 0$. From the picture, $d=0$ when $theta =0$.
– Matthew Leingang
Jul 31 at 18:17
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to figure out what is wrong with my solution to the following, posted below
Suppose, without loss of generality that the circle has radius $1$. Then $theta = s$. From the law of cosines, we have $d=2+2cos(theta)$. From this we get that
$$lim_thetarightarrow 0fracsd=lim_thetarightarrow 0fractheta2-2cos(theta)=lim_thetarightarrow 0frac12sin(theta)=infty.
$$
However, I am told that the answer is $1$. What is wrong with my reasoning?
calculus
edited Jul 31 at 18:18
gt6989b
30.2k22148
asked Jul 31 at 18:14
ponchan
34719
I am trying to figure out what is wrong with my solution to the following, posted below
Suppose, without loss of generality that the circle has radius $1$. Then $theta = s$. From the law of cosines, we have $d=2+2cos(theta)$. From this we get that
$$lim_thetarightarrow 0fracsd=lim_thetarightarrow 0fractheta2-2cos(theta)=lim_thetarightarrow 0frac12sin(theta)=infty.
$$
However, I am told that the answer is $1$. What is wrong with my reasoning?
calculus
edited Jul 31 at 18:18
gt6989b
30.2k22148
asked Jul 31 at 18:14
ponchan
34719
edited Jul 31 at 18:18
gt6989b
30.2k22148
edited Jul 31 at 18:18
gt6989b
30.2k22148
edited Jul 31 at 18:18
gt6989b
30.2k22148
30.2k22148
asked Jul 31 at 18:14
ponchan
34719
asked Jul 31 at 18:14
ponchan
34719
asked Jul 31 at 18:14
ponchan
34719
34719
3
$d=2+2costheta$ is certainly wrong, since that predicts $d=4$ when $theta = 0$. From the picture, $d=0$ when $theta =0$.
– Matthew Leingang
Jul 31 at 18:17
add a comment |Â
3
$d=2+2costheta$ is certainly wrong, since that predicts $d=4$ when $theta = 0$. From the picture, $d=0$ when $theta =0$.
– Matthew Leingang
Jul 31 at 18:17
3
3
$d=2+2costheta$ is certainly wrong, since that predicts $d=4$ when $theta = 0$. From the picture, $d=0$ when $theta =0$.
– Matthew Leingang
Jul 31 at 18:17
$d=2+2costheta$ is certainly wrong, since that predicts $d=4$ when $theta = 0$. From the picture, $d=0$ when $theta =0$.
– Matthew Leingang
Jul 31 at 18:17
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
3
down vote
accepted
Isn't it $$d^2=1+1-2cos(theta)$$?
and use that
$$cos(theta)=1-2sin^2left(fractheta2right)$$
answered Jul 31 at 18:17
Dr. Sonnhard Graubner
66.6k32659
add a comment |Â
up vote
1
down vote
We have isosceles triangle with legs equal to $1$ and vertex angle equal to $theta$. Thus, $$d=2 cdot sin(fractheta2)$$
answered Jul 31 at 18:31
Vasya
2,4701514
add a comment |Â
up vote
0
down vote
If you devide the triangle you find that :
$$d=2 cdot sin(fractheta2)$$
$$s=theta$$
thus
$$fracsd=fractheta2 cdot sin(fractheta2)$$
$$fracsd =fracfractheta2sin(fractheta2)$$
finally
$$lim_thetarightarrow 0fracsd=lim_thetarightarrow 0fracfractheta2sin(fractheta2)=1$$
answered Jul 31 at 18:39
Gog Magog
1
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Isn't it $$d^2=1+1-2cos(theta)$$?
and use that
$$cos(theta)=1-2sin^2left(fractheta2right)$$
answered Jul 31 at 18:17
Dr. Sonnhard Graubner
66.6k32659
add a comment |Â
up vote
3
down vote
accepted
Isn't it $$d^2=1+1-2cos(theta)$$?
and use that
$$cos(theta)=1-2sin^2left(fractheta2right)$$
answered Jul 31 at 18:17
Dr. Sonnhard Graubner
66.6k32659
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Isn't it $$d^2=1+1-2cos(theta)$$?
and use that
$$cos(theta)=1-2sin^2left(fractheta2right)$$
answered Jul 31 at 18:17
Dr. Sonnhard Graubner
66.6k32659
Isn't it $$d^2=1+1-2cos(theta)$$?
and use that
$$cos(theta)=1-2sin^2left(fractheta2right)$$
answered Jul 31 at 18:17
Dr. Sonnhard Graubner
66.6k32659
answered Jul 31 at 18:17
Dr. Sonnhard Graubner
66.6k32659
answered Jul 31 at 18:17
Dr. Sonnhard Graubner
66.6k32659
answered Jul 31 at 18:17
Dr. Sonnhard Graubner
66.6k32659
66.6k32659
add a comment |Â
add a comment |Â
up vote
1
down vote
We have isosceles triangle with legs equal to $1$ and vertex angle equal to $theta$. Thus, $$d=2 cdot sin(fractheta2)$$
answered Jul 31 at 18:31
Vasya
2,4701514
add a comment |Â
up vote
1
down vote
We have isosceles triangle with legs equal to $1$ and vertex angle equal to $theta$. Thus, $$d=2 cdot sin(fractheta2)$$
answered Jul 31 at 18:31
Vasya
2,4701514
add a comment |Â
up vote
1
down vote
up vote
1
down vote
We have isosceles triangle with legs equal to $1$ and vertex angle equal to $theta$. Thus, $$d=2 cdot sin(fractheta2)$$
answered Jul 31 at 18:31
Vasya
2,4701514
We have isosceles triangle with legs equal to $1$ and vertex angle equal to $theta$. Thus, $$d=2 cdot sin(fractheta2)$$
answered Jul 31 at 18:31
Vasya
2,4701514
answered Jul 31 at 18:31
Vasya
2,4701514
answered Jul 31 at 18:31
Vasya
2,4701514
answered Jul 31 at 18:31
Vasya
2,4701514
2,4701514
add a comment |Â
add a comment |Â
up vote
0
down vote
If you devide the triangle you find that :
$$d=2 cdot sin(fractheta2)$$
$$s=theta$$
thus
$$fracsd=fractheta2 cdot sin(fractheta2)$$
$$fracsd =fracfractheta2sin(fractheta2)$$
finally
$$lim_thetarightarrow 0fracsd=lim_thetarightarrow 0fracfractheta2sin(fractheta2)=1$$
answered Jul 31 at 18:39
Gog Magog
1
add a comment |Â
up vote
0
down vote
If you devide the triangle you find that :
$$d=2 cdot sin(fractheta2)$$
$$s=theta$$
thus
$$fracsd=fractheta2 cdot sin(fractheta2)$$
$$fracsd =fracfractheta2sin(fractheta2)$$
finally
$$lim_thetarightarrow 0fracsd=lim_thetarightarrow 0fracfractheta2sin(fractheta2)=1$$
answered Jul 31 at 18:39
Gog Magog
1
add a comment |Â
up vote
0
down vote
up vote
0
down vote
If you devide the triangle you find that :
$$d=2 cdot sin(fractheta2)$$
$$s=theta$$
thus
$$fracsd=fractheta2 cdot sin(fractheta2)$$
$$fracsd =fracfractheta2sin(fractheta2)$$
finally
$$lim_thetarightarrow 0fracsd=lim_thetarightarrow 0fracfractheta2sin(fractheta2)=1$$
answered Jul 31 at 18:39
Gog Magog
1
If you devide the triangle you find that :
$$d=2 cdot sin(fractheta2)$$
$$s=theta$$
thus
$$fracsd=fractheta2 cdot sin(fractheta2)$$
$$fracsd =fracfractheta2sin(fractheta2)$$
finally
$$lim_thetarightarrow 0fracsd=lim_thetarightarrow 0fracfractheta2sin(fractheta2)=1$$
answered Jul 31 at 18:39
Gog Magog
1
answered Jul 31 at 18:39
Gog Magog
1
answered Jul 31 at 18:39
Gog Magog
1
answered Jul 31 at 18:39
Gog Magog
1
1
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%2f2868321%2flimit-involving-trig%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
3
$d=2+2costheta$ is certainly wrong, since that predicts $d=4$ when $theta = 0$. From the picture, $d=0$ when $theta =0$.
– Matthew Leingang
Jul 31 at 18:17