trigonometry: what's the range of cosα sinβ if sinα cosβ = 1/2 ?

hearts123

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Feb 22, 2019
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Hi everyone, I need some help with this question.
I'm given: sin α cos β = 1/2
what's the range of cos α sin β ?
I think I'm supposed to use identities to get the answer.
sin(α + β) = sin α cos β + cos α sin β
sin(α + β) = 1/2 + cos α sin β
After that, I don't know what to do. Should I isolate the cos α sin β on one side of the equal sign, or should I isolate the 1/2? Or is there another identity I can use?
Any help is appreciated!
 

tkhunny

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Hi everyone, I need some help with this question.
I'm given: sin α cos β = 1/2
what's the range of cos α sin β ?
I think I'm supposed to use identities to get the answer.
sin(α + β) = sin α cos β + cos α sin β
sin(α + β) = 1/2 + cos α sin β
After that, I don't know what to do. Should I isolate the cos α sin β on one side of the equal sign, or should I isolate the 1/2? Or is there another identity I can use?
Any help is appreciated!
I'm not super clear on what is being asked.

Possible Hint: \(\displaystyle -1\le\sin\left(\alpha + \beta\right)\le 1\)
 

hearts123

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Feb 22, 2019
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I'm not super clear on what is being asked.

Possible Hint: \(\displaystyle -1\le\sin\left(\alpha + \beta\right)\le 1\)
The question asks for the range of cos α sin β as in, what's the minimum possible value for cos α sin β and what's the maximum possible value for it?
Your hint is just what I needed to get the answer, thank you so much :)
(hopefully I got this right)
The minimum would be -1/2, the maximum would be 1/2
Substituting -1 as sin(α + β) and moving the 1/2 over to the other side to get -1/2
Substituting 1 as sin(α + β) and moving 1/2 to the other side to get 1/2 as the maximum

How did you find/figure out the hint though? Is it from sin α cos β = 1/2 ?
 

ksdhart2

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How did you find/figure out the hint though? Is it from sin α cos β = 1/2 ?

It's a known fact of the sine function that \(\displaystyle \forall x \in \mathbb{R}: -1 \le sin(x) \le 1\) (i.e. this property holds so long as x is a real number). Let \(\displaystyle x = \alpha + \beta\) and Bob's your uncle.
 

Dr.Peterson

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I think there's one piece missing: Do you know that \(\displaystyle \alpha+\beta\) can take every possible value, under the stated condition?

From sketching the graph of the condition, I think the answer is yes. But you need to prove that (or something a little less).
 

tkhunny

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I'm not super clear on what is being asked.

Possible Hint: \(\displaystyle -1\le\sin\left(\alpha + \beta\right)\le 1\)
It appears I was close?

Extended Hint: \(\displaystyle -1\le 1/2 + \cos(\alpha)\cdot\sin(\beta)\le 1\)
 

MarkFL

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