Given $sin(x)$, find $sin(frac{x}{2}) cos(frac{5x}{2})$ [closed]

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Knowing that
$pi < 2x < 2pi$
and

$$sin(x) = frac{4}{5},$$

find

$$sinleft(frac{x}{2}right) cosleft(frac{5x}{2}right) = ?$$

edited Nov 22 at 14:13

Tianlalu

2,859832

asked Nov 22 at 14:07

critical_mass

closed as off-topic by Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo Nov 23 at 1:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.

1

mathworld.wolfram.com/WernerFormulas.html
– lab bhattacharjee
Nov 22 at 14:17

add a comment |

up vote
-1
down vote

favorite

Knowing that
$pi < 2x < 2pi$
and

$$sin(x) = frac{4}{5},$$

find

$$sinleft(frac{x}{2}right) cosleft(frac{5x}{2}right) = ?$$

edited Nov 22 at 14:13

Tianlalu

2,859832

asked Nov 22 at 14:07

critical_mass

closed as off-topic by Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo Nov 23 at 1:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.

1

mathworld.wolfram.com/WernerFormulas.html
– lab bhattacharjee
Nov 22 at 14:17

add a comment |

up vote
-1
down vote

favorite

Knowing that
$pi < 2x < 2pi$
and

$$sin(x) = frac{4}{5},$$

find

$$sinleft(frac{x}{2}right) cosleft(frac{5x}{2}right) = ?$$

edited Nov 22 at 14:13

Tianlalu

2,859832

asked Nov 22 at 14:07

critical_mass

Knowing that
$pi < 2x < 2pi$
and

$$sin(x) = frac{4}{5},$$

find

$$sinleft(frac{x}{2}right) cosleft(frac{5x}{2}right) = ?$$

trigonometry transformation

edited Nov 22 at 14:13

Tianlalu

2,859832

asked Nov 22 at 14:07

critical_mass

edited Nov 22 at 14:13

Tianlalu

2,859832

asked Nov 22 at 14:07

critical_mass

edited Nov 22 at 14:13

Tianlalu

2,859832

edited Nov 22 at 14:13

Tianlalu

2,859832

edited Nov 22 at 14:13

Tianlalu

2,859832

asked Nov 22 at 14:07

critical_mass

asked Nov 22 at 14:07

critical_mass

asked Nov 22 at 14:07

critical_mass

closed as off-topic by Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo Nov 23 at 1:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo Nov 23 at 1:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.

1

mathworld.wolfram.com/WernerFormulas.html
– lab bhattacharjee
Nov 22 at 14:17

add a comment |

1

mathworld.wolfram.com/WernerFormulas.html
– lab bhattacharjee
Nov 22 at 14:17

mathworld.wolfram.com/WernerFormulas.html
– lab bhattacharjee
Nov 22 at 14:17

add a comment |

2 Answers
2

active

oldest

votes

up vote
4
down vote

accepted

HINT

Use that

$$sin theta cos varphi = frac12{{sin(theta + varphi) + frac12 sin(theta - varphi)} }$$

and

$sin (2theta) =2sintheta costheta$

$sin (3theta) =3sintheta - 4sin^3theta$

moreover from the given

$cos x=-sqrt{1-sin^2 x}$

edited Nov 22 at 14:13

answered Nov 22 at 14:10

gimusi

88.8k74394

add a comment |

up vote
2
down vote

$sin(x/2)cos(5x/2)\
=sin(x/2)cos(2x+x/2)\=sin(x/2){cos(2x)cos(x/2)-sin(2x)sin(x/2)}\=cos(2x)cos(x/2)sin(x/2)-sin(2x)sin^2(x/2)\=1/2(1-2sin^2(x))sin(x)-sin(x)cos(x)(1-cos(x))$

As $pi/2<x<pi$, $sin(x)=4/5$ implies $cos(x)=-3/5$. So the final answer is $sin(x/2)cos(5x/2)=82/125.$

answered Nov 22 at 14:23

John_Wick

84919

Welcome to to MathSE. Type $sin x$ , $cos x$ , $tan x$ , $csc x$ , $sec x$ , and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
– N. F. Taussig
Nov 22 at 14:34

add a comment |

2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

up vote
4
down vote

accepted

HINT

Use that

$$sin theta cos varphi = frac12{{sin(theta + varphi) + frac12 sin(theta - varphi)} }$$

and

$sin (2theta) =2sintheta costheta$

$sin (3theta) =3sintheta - 4sin^3theta$

moreover from the given

$cos x=-sqrt{1-sin^2 x}$

edited Nov 22 at 14:13

answered Nov 22 at 14:10

gimusi

88.8k74394

add a comment |

up vote
4
down vote

accepted

HINT

Use that

$$sin theta cos varphi = frac12{{sin(theta + varphi) + frac12 sin(theta - varphi)} }$$

and

$sin (2theta) =2sintheta costheta$

$sin (3theta) =3sintheta - 4sin^3theta$

moreover from the given

$cos x=-sqrt{1-sin^2 x}$

edited Nov 22 at 14:13

answered Nov 22 at 14:10

gimusi

88.8k74394

add a comment |

up vote
4
down vote

accepted

HINT

Use that

$$sin theta cos varphi = frac12{{sin(theta + varphi) + frac12 sin(theta - varphi)} }$$

and

$sin (2theta) =2sintheta costheta$

$sin (3theta) =3sintheta - 4sin^3theta$

moreover from the given

$cos x=-sqrt{1-sin^2 x}$

edited Nov 22 at 14:13

answered Nov 22 at 14:10

gimusi

88.8k74394

HINT

Use that

$$sin theta cos varphi = frac12{{sin(theta + varphi) + frac12 sin(theta - varphi)} }$$

and

$sin (2theta) =2sintheta costheta$

$sin (3theta) =3sintheta - 4sin^3theta$

moreover from the given

$cos x=-sqrt{1-sin^2 x}$

edited Nov 22 at 14:13

answered Nov 22 at 14:10

gimusi

88.8k74394

edited Nov 22 at 14:13

answered Nov 22 at 14:10

gimusi

88.8k74394

answered Nov 22 at 14:10

gimusi

88.8k74394

answered Nov 22 at 14:10

gimusi

88.8k74394

add a comment |

up vote
2
down vote

$sin(x/2)cos(5x/2)\
=sin(x/2)cos(2x+x/2)\=sin(x/2){cos(2x)cos(x/2)-sin(2x)sin(x/2)}\=cos(2x)cos(x/2)sin(x/2)-sin(2x)sin^2(x/2)\=1/2(1-2sin^2(x))sin(x)-sin(x)cos(x)(1-cos(x))$

As $pi/2<x<pi$, $sin(x)=4/5$ implies $cos(x)=-3/5$. So the final answer is $sin(x/2)cos(5x/2)=82/125.$

answered Nov 22 at 14:23

John_Wick

84919

Welcome to to MathSE. Type $sin x$ , $cos x$ , $tan x$ , $csc x$ , $sec x$ , and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
– N. F. Taussig
Nov 22 at 14:34

add a comment |

up vote
2
down vote

$sin(x/2)cos(5x/2)\
=sin(x/2)cos(2x+x/2)\=sin(x/2){cos(2x)cos(x/2)-sin(2x)sin(x/2)}\=cos(2x)cos(x/2)sin(x/2)-sin(2x)sin^2(x/2)\=1/2(1-2sin^2(x))sin(x)-sin(x)cos(x)(1-cos(x))$

As $pi/2<x<pi$, $sin(x)=4/5$ implies $cos(x)=-3/5$. So the final answer is $sin(x/2)cos(5x/2)=82/125.$

answered Nov 22 at 14:23

John_Wick

84919

Welcome to to MathSE. Type $sin x$ , $cos x$ , $tan x$ , $csc x$ , $sec x$ , and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
– N. F. Taussig
Nov 22 at 14:34

add a comment |

up vote
2
down vote

$sin(x/2)cos(5x/2)\
=sin(x/2)cos(2x+x/2)\=sin(x/2){cos(2x)cos(x/2)-sin(2x)sin(x/2)}\=cos(2x)cos(x/2)sin(x/2)-sin(2x)sin^2(x/2)\=1/2(1-2sin^2(x))sin(x)-sin(x)cos(x)(1-cos(x))$

As $pi/2<x<pi$, $sin(x)=4/5$ implies $cos(x)=-3/5$. So the final answer is $sin(x/2)cos(5x/2)=82/125.$

answered Nov 22 at 14:23

John_Wick

84919

$sin(x/2)cos(5x/2)\
=sin(x/2)cos(2x+x/2)\=sin(x/2){cos(2x)cos(x/2)-sin(2x)sin(x/2)}\=cos(2x)cos(x/2)sin(x/2)-sin(2x)sin^2(x/2)\=1/2(1-2sin^2(x))sin(x)-sin(x)cos(x)(1-cos(x))$

As $pi/2<x<pi$, $sin(x)=4/5$ implies $cos(x)=-3/5$. So the final answer is $sin(x/2)cos(5x/2)=82/125.$

answered Nov 22 at 14:23

John_Wick

84919

answered Nov 22 at 14:23

John_Wick

84919

answered Nov 22 at 14:23

John_Wick

84919

answered Nov 22 at 14:23

John_Wick

84919

Welcome to to MathSE. Type $sin x$ , $cos x$ , $tan x$ , $csc x$ , $sec x$ , and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
– N. F. Taussig
Nov 22 at 14:34

add a comment |

Welcome to to MathSE. Type $sin x$ , $cos x$ , $tan x$ , $csc x$ , $sec x$ , and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
– N. F. Taussig
Nov 22 at 14:34

Welcome to to MathSE. Type $sin x$ , $cos x$ , $tan x$ , $csc x$ , $sec x$ , and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
– N. F. Taussig
Nov 22 at 14:34

add a comment |

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