savac
Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle.
cos θ = − 5/9, sin θ > 0
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PeterPan
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Okay, let's make that drawing again, though following @hartnn 's advice, I'll do it properly this time ;)
DRAW!!!
|dw:1360757734426:dw|
PeterPan
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Cosine is adj/hyp, right?
savac
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yes
PeterPan
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So, we have cosθ = -(5/9)
We place the 5 and the 9, accordingly...
|dw:1360757883210:dw|
(Stop me when you don't understand)
savac
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i understand
PeterPan
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Now, what's the length of this missing side... |dw:1360758005338:dw|
(Hint: Pythagorean theorem)
savac
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\[\sqrt{106}\] right?
PeterPan
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Remember, 9 is your hypotenuse
When you have
a^2 + b^2 = c^2
9 is your c :)
mathsmind
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\[\cos ^2(\theta)+\sin^2(\theta)=1\]
mathsmind
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\[\sin^2(\theta) = 1-\cos^2(\theta)\]
savac
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\[\sqrt{56}\]
mathsmind
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\[\sin^2(\theta)=(\frac{- 5 }{ 9 })^2 -1\]
mathsmind
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i prefer the method provided by @PeterPan, because it shows you the inside out
PeterPan
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So, we can put sqrt(56) on here...|dw:1360758287438:dw|
And you can easily get the other trig functions. Be careful though...
PeterPan
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@mathsmind
You do need to do this to illustrate the pythagorean identities you posted. Though if you're not allowed to draw, better know the pythagorean (and other) identities by heart!!!
:D
mathsmind
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yes that is what i said
mathsmind
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i said i prefer the other method to know the inside out
savac
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we are allowed to draw on the test
PeterPan
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@savac
REMEMBER THIS
|dw:1360758381571:dw|
mathsmind
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although the identity was proven using pyth.
savac
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whats that?
PeterPan
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ACTS!!!!
|dw:1360758433801:dw|
savac
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ohhhh thats smart
mathsmind
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savac PeterPan is doing a great job
PeterPan
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@mathsmind
Thanks for the kudos!
Much appreciated :)
PeterPan
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So back to this...
|dw:1360758603756:dw|
Can you now determine the rest of the trig functions of this theta?
savac
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give me a mint
savac
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min
PeterPan
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:D
I was about to say something clever, but you beat me to it :)
PeterPan
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REMEMBER
In this particular quadrant, only sine (and cosecant) are positive
savac
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lol sorry running low on sleep
savac
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ok i got sin=\[\sqrt{56}\]/9
tan=-\[\sqrt{56}\]/5
csc=9/\[\sqrt{56}\]
sec=-9/5
cot=-5/\[\sqrt{56}\]
PeterPan
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What about cos?
savac
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cos=-5/9
PeterPan
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And there you have it :)
savac
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that was rright?
PeterPan
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Well yeah, unless you want to make it
\[\huge \sqrt{56}=2\sqrt{14}\]
savac
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wow i surprised myself. lol thank you very much
PeterPan
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Just place that right triangle in the correct quadrant, and you can't go wrong :)
You'll notice that the horizontal bit was to the left of the x-axis, so strictly speaking, that's a -5, not a 5 :D
savac
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thats what i thought it was
hartnn
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actually since theta is in 2nd quadrant, that is its measure is between 90 to 180, the triangle will be , |dw:1360722028091:dw|
PeterPan
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Looks more complicated.
hartnn
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but the way its solved and answers are correct.
hartnn
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sin and cosec will be positive, all other ratios will be negative.
PeterPan
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Yay :)
hartnn
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if you want to solve using diagram, then what i have drawn must be used .
here, its solved using identities, which is altogether different method and doesn't need drawing triangle.
PeterPan
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Identities are harder to swallow, and they taste kinda awful too :(