Mixing
Show that when written in terms of $ t$, where $t = tan(x/2)$, the expression $2(1 + cos(x))(5sin(x) +...
$begingroup$
Attempt:
I've used that $sin(x) = (2t)/(1+t^2)$ and $cos(x) = (1-t^2)/(1+t^2)$.
However I don't seem to get a perfect square, instead I get $$(2/((1+t^2)^2))(14t^2 +20t + 38)$$.
I'm not sure if there error is with my method or my workings.
Any help would be greatly appreciated.
trigonometry
edited Jan 25 at 11:30
Larry
2,53531131
asked Jan 25 at 11:12
B.GriffinB.Griffin
113
$endgroup$
add a comment |
$begingroup$
Attempt:
I've used that $sin(x) = (2t)/(1+t^2)$ and $cos(x) = (1-t^2)/(1+t^2)$.
However I don't seem to get a perfect square, instead I get $$(2/((1+t^2)^2))(14t^2 +20t + 38)$$.
I'm not sure if there error is with my method or my workings.
Any help would be greatly appreciated.
trigonometry
edited Jan 25 at 11:30
Larry
2,53531131
asked Jan 25 at 11:12
B.GriffinB.Griffin
113
$endgroup$
add a comment |
$begingroup$
Attempt:
I've used that $sin(x) = (2t)/(1+t^2)$ and $cos(x) = (1-t^2)/(1+t^2)$.
However I don't seem to get a perfect square, instead I get $$(2/((1+t^2)^2))(14t^2 +20t + 38)$$.
I'm not sure if there error is with my method or my workings.
Any help would be greatly appreciated.
trigonometry
edited Jan 25 at 11:30
Larry
2,53531131
asked Jan 25 at 11:12
B.GriffinB.Griffin
113
$endgroup$
Attempt:
I've used that $sin(x) = (2t)/(1+t^2)$ and $cos(x) = (1-t^2)/(1+t^2)$.
However I don't seem to get a perfect square, instead I get $$(2/((1+t^2)^2))(14t^2 +20t + 38)$$.
I'm not sure if there error is with my method or my workings.
Any help would be greatly appreciated.
trigonometry
trigonometry
edited Jan 25 at 11:30
Larry
2,53531131
asked Jan 25 at 11:12
B.GriffinB.Griffin
113
edited Jan 25 at 11:30
Larry
2,53531131
asked Jan 25 at 11:12
B.GriffinB.Griffin
113
edited Jan 25 at 11:30
Larry
2,53531131
edited Jan 25 at 11:30
Larry
2,53531131
edited Jan 25 at 11:30
Larry
2,53531131
2,53531131
asked Jan 25 at 11:12
B.GriffinB.Griffin
113
asked Jan 25 at 11:12
B.GriffinB.Griffin
113
asked Jan 25 at 11:12
B.GriffinB.Griffin
113
113
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
$$begin{align}
2(1+cos(x))(5sin(x)+12cos(x)+13)&=2left(1+frac{1-t^2}{1+t^2}right)left(frac{10t}{1+t^2}+frac{12-12t^2}{1+t^2}+13right)\
&=frac{4}{(1+t^2)^2}(t+5)^2
end{align}$$
edited Jan 25 at 11:33
Larry
2,53531131
answered Jan 25 at 11:25
J.DaneJ.Dane
370114
$endgroup$
add a comment |
$begingroup$
I get
$$frac{4 {{left( t+5right) }^{2}}}{{{left( {{t}^{2}}+1right) }^{2}}}$$
answered Jan 25 at 11:24
Aleksas DomarkasAleksas Domarkas
1,53816
$endgroup$
add a comment |
$begingroup$
If $x= pi/4$, then compute that
$2(1+ cos x)(5 sin x+ 12 cos x+13)$ is not a natural number !
answered Jan 25 at 11:25
FredFred
48.5k11849
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$$begin{align}
2(1+cos(x))(5sin(x)+12cos(x)+13)&=2left(1+frac{1-t^2}{1+t^2}right)left(frac{10t}{1+t^2}+frac{12-12t^2}{1+t^2}+13right)\
&=frac{4}{(1+t^2)^2}(t+5)^2
end{align}$$
edited Jan 25 at 11:33
Larry
2,53531131
answered Jan 25 at 11:25
J.DaneJ.Dane
370114
$endgroup$
add a comment |
$begingroup$
$$begin{align}
2(1+cos(x))(5sin(x)+12cos(x)+13)&=2left(1+frac{1-t^2}{1+t^2}right)left(frac{10t}{1+t^2}+frac{12-12t^2}{1+t^2}+13right)\
&=frac{4}{(1+t^2)^2}(t+5)^2
end{align}$$
edited Jan 25 at 11:33
Larry
2,53531131
answered Jan 25 at 11:25
J.DaneJ.Dane
370114
$endgroup$
add a comment |
$begingroup$
$$begin{align}
2(1+cos(x))(5sin(x)+12cos(x)+13)&=2left(1+frac{1-t^2}{1+t^2}right)left(frac{10t}{1+t^2}+frac{12-12t^2}{1+t^2}+13right)\
&=frac{4}{(1+t^2)^2}(t+5)^2
end{align}$$
edited Jan 25 at 11:33
Larry
2,53531131
answered Jan 25 at 11:25
J.DaneJ.Dane
370114
$endgroup$
$$begin{align}
2(1+cos(x))(5sin(x)+12cos(x)+13)&=2left(1+frac{1-t^2}{1+t^2}right)left(frac{10t}{1+t^2}+frac{12-12t^2}{1+t^2}+13right)\
&=frac{4}{(1+t^2)^2}(t+5)^2
end{align}$$
edited Jan 25 at 11:33
Larry
2,53531131
answered Jan 25 at 11:25
J.DaneJ.Dane
370114
edited Jan 25 at 11:33
Larry
2,53531131
edited Jan 25 at 11:33
Larry
2,53531131
edited Jan 25 at 11:33
Larry
2,53531131
2,53531131
answered Jan 25 at 11:25
J.DaneJ.Dane
370114
answered Jan 25 at 11:25
J.DaneJ.Dane
370114
answered Jan 25 at 11:25
J.DaneJ.Dane
370114
370114
add a comment |
add a comment |
$begingroup$
I get
$$frac{4 {{left( t+5right) }^{2}}}{{{left( {{t}^{2}}+1right) }^{2}}}$$
answered Jan 25 at 11:24
Aleksas DomarkasAleksas Domarkas
1,53816
$endgroup$
add a comment |
$begingroup$
I get
$$frac{4 {{left( t+5right) }^{2}}}{{{left( {{t}^{2}}+1right) }^{2}}}$$
answered Jan 25 at 11:24
Aleksas DomarkasAleksas Domarkas
1,53816
$endgroup$
add a comment |
$begingroup$
I get
$$frac{4 {{left( t+5right) }^{2}}}{{{left( {{t}^{2}}+1right) }^{2}}}$$
answered Jan 25 at 11:24
Aleksas DomarkasAleksas Domarkas
1,53816
$endgroup$
I get
$$frac{4 {{left( t+5right) }^{2}}}{{{left( {{t}^{2}}+1right) }^{2}}}$$
answered Jan 25 at 11:24
Aleksas DomarkasAleksas Domarkas
1,53816
answered Jan 25 at 11:24
Aleksas DomarkasAleksas Domarkas
1,53816
answered Jan 25 at 11:24
Aleksas DomarkasAleksas Domarkas
1,53816
answered Jan 25 at 11:24
Aleksas DomarkasAleksas Domarkas
1,53816
1,53816
add a comment |
add a comment |
$begingroup$
If $x= pi/4$, then compute that
$2(1+ cos x)(5 sin x+ 12 cos x+13)$ is not a natural number !
answered Jan 25 at 11:25
FredFred
48.5k11849
$endgroup$
add a comment |
$begingroup$
If $x= pi/4$, then compute that
$2(1+ cos x)(5 sin x+ 12 cos x+13)$ is not a natural number !
answered Jan 25 at 11:25
FredFred
48.5k11849
$endgroup$
add a comment |
$begingroup$
If $x= pi/4$, then compute that
$2(1+ cos x)(5 sin x+ 12 cos x+13)$ is not a natural number !
answered Jan 25 at 11:25
FredFred
48.5k11849
$endgroup$
If $x= pi/4$, then compute that
$2(1+ cos x)(5 sin x+ 12 cos x+13)$ is not a natural number !
answered Jan 25 at 11:25
FredFred
48.5k11849
answered Jan 25 at 11:25
FredFred
48.5k11849
answered Jan 25 at 11:25
FredFred
48.5k11849
answered Jan 25 at 11:25
FredFred
48.5k11849
48.5k11849
add a comment |
add a comment |
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