Solution 4

Intro. To Problem 4Intro. To Problem 4

You are very tired. You are doing a project that is deterring your efforts to study You are very tired. You are doing a project that is deterring your efforts to study for exams. You feel that you will not be prepared for exams because of that for exams. You feel that you will not be prepared for exams because of that certain project. Today is the due date of your project and it is 3:25 AM. You still certain project. Today is the due date of your project and it is 3:25 AM. You still need to think of one question. Then after that, you’re going to need to provide need to think of one question. Then after that, you’re going to need to provide an explanation for it and post it on your website to showcase your project. You an explanation for it and post it on your website to showcase your project. You keep thinking to yourself, “What should the final question be?” You come to keep thinking to yourself, “What should the final question be?” You come to understand that your mind has been weakened from lack of sleep, stress, and understand that your mind has been weakened from lack of sleep, stress, and too much partying. Even great minds can’t always think.too much partying. Even great minds can’t always think.

What you do is cease your fruitless efforts on your project and review for math What you do is cease your fruitless efforts on your project and review for math because you understand that the provincial math exam is approaching in about because you understand that the provincial math exam is approaching in about 4 days. You grab some sheets and begin reading.4 days. You grab some sheets and begin reading.

You look at one certain question. Just by looking at it, you freeze up from not You look at one certain question. Just by looking at it, you freeze up from not knowing how to answer it. knowing how to answer it.

Well well, let’s take a look at that question again… Muahaha!!!Well well, let’s take a look at that question again… Muahaha!!!

Elegant ProblemElegant Problem

Determine Theta Given The Equation Below:Determine Theta Given The Equation Below:

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This is the last question, which is the most simple looking of the four, yet This is the last question, which is the most simple looking of the four, yet the question itself requires lots of thinking.the question itself requires lots of thinking.

Let’s start by rearranging the equation. After the rearrangement, you Let’s start by rearranging the equation. After the rearrangement, you should notice something.should notice something.

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Maybe the brackets help you notice this thing. We can in fact factor the Maybe the brackets help you notice this thing. We can in fact factor the portion in brackets out.portion in brackets out.

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You should notice another thing as well. The Pythagorean Identity is You should notice another thing as well. The Pythagorean Identity is apparent.apparent.



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Sin^2Θ + Cos^2Θ is equal to 1. So now we just have this.Sin^2Θ + Cos^2Θ is equal to 1. So now we just have this.

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Cos^2Θ – Cos^2Θ equal zero, so you are just left with Sin^2Θ.Cos^2Θ – Cos^2Θ equal zero, so you are just left with Sin^2Θ.



=Now we obtain the reciprocal of the reciprocal form of sin^2Θ, which is Now we obtain the reciprocal of the reciprocal form of sin^2Θ, which is 1/csc^2Θ. This value is equal to sin^2Θ. Cosecant is the reciprocal of 1/csc^2Θ. This value is equal to sin^2Θ. Cosecant is the reciprocal of

Sine, so if we take the reciprocal of Cosecant, we obtain Sine.Sine, so if we take the reciprocal of Cosecant, we obtain Sine.

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You should notice that we can isolate 1.You should notice that we can isolate 1.



=Hmmm… All the stuff on the left must equal 1. Maybe this will help you Hmmm… All the stuff on the left must equal 1. Maybe this will help you

understand what to do. I will assign cscΘ a letter.understand what to do. I will assign cscΘ a letter.Let Q=cscΘLet Q=cscΘ

=What value for Q squared times Q to the exponent Q to the exponent Q will give What value for Q squared times Q to the exponent Q to the exponent Q will give

1? Well, if you think about it, The only value that satisfies this equation is 1.1? Well, if you think about it, The only value that satisfies this equation is 1.1^2=1 1^1=11^2=1 1^1=1

The values are always one, so cscΘ must equal 1.The values are always one, so cscΘ must equal 1.



=Now we can change cosecant into terms of sin, so that we can use our Now we can change cosecant into terms of sin, so that we can use our

heads or calculator to solve for theta.heads or calculator to solve for theta.

Now we must isolate sinΘ to obtain its exact value. We can do all of this to isolate sinΘ:Now we must isolate sinΘ to obtain its exact value. We can do all of this to isolate sinΘ:Obtain the reciprocal of both sidesObtain the reciprocal of both sidesCross multiplyCross multiplyMultiply both sides by sinΘMultiply both sides by sinΘDivide both sides by 1/sinΘDivide both sides by 1/sinΘ

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In the end, sinΘ=1. You can use your calculator for this, but you really should not have to. If In the end, sinΘ=1. You can use your calculator for this, but you really should not have to. If you do, use the arcsine function.you do, use the arcsine function.

It is obvious that theta = 90 degrees or Π/2 radiansIt is obvious that theta = 90 degrees or Π/2 radians

Now let’s take a look at a unit circle to see this.Now let’s take a look at a unit circle to see this.



Now the question did not indicate an interval for theta, so let’s just assume that it wants every Now the question did not indicate an interval for theta, so let’s just assume that it wants every angle for when sinΘ=1.angle for when sinΘ=1.

DegreesDegreesΘ = 90° + K360 where K is an element of IΘ = 90° + K360 where K is an element of I

AndAnd

RadiansRadiansΘ = Π/2 + K2Π where K is an element of IΘ = Π/2 + K2Π where K is an element of I

Congratulations! You’re Congratulations! You’re Done!Done!

Date post:	19-Jun-2015
Category:	Technology
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Solution 4

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