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# Solution 4

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### Solution 4

1. 1. Intro. To Problem 4 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 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 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 understand that your mind has been weakened from lack of sleep, stress, and 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 because you understand that the provincial math exam is approaching in about 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 knowing how to answer it. Well well, let’s take a look at that question again… Muahaha!!!
2. 2. Elegant Problem Determine Theta Given The Equation Below: =
3. 3. Elegant Problem Determine Theta Given The Equation Below: = This is the last question, which is the most simple looking of the four, yet the question itself requires lots of thinking. Let’s start by rearranging the equation. After the rearrangement, you should notice something. =
4. 4. Elegant Problem Determine Theta Given The Equation Below: = Maybe the brackets help you notice this thing. We can in fact factor the portion in brackets out. = You should notice another thing as well. The Pythagorean Identity is apparent.
5. 5. Elegant Problem Determine Theta Given The Equation Below: = Sin^2Θ + Cos^2Θ is equal to 1. So now we just have this. = Cos^2Θ – Cos^2Θ equal zero, so you are just left with Sin^2Θ.
6. 6. Elegant Problem Determine Theta Given The Equation Below: = 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 Sine, so if we take the reciprocal of Cosecant, we obtain Sine. = You should notice that we can isolate 1.
7. 7. Elegant Problem Determine Theta Given The Equation Below: = 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. Let Q=cscΘ = 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. 1^2=1 1^1=1 The values are always one, so cscΘ must equal 1.
8. 8. Elegant Problem Determine Theta Given The Equation Below: = Now we can change cosecant into terms of sin, so that we can use our heads or calculator to solve for theta. <ul><li>Now we must isolate sinΘ to obtain its exact value. We can do all of this to isolate sinΘ: </li></ul><ul><li>Obtain the reciprocal of both sides </li></ul><ul><li>Cross multiply </li></ul><ul><li>Multiply both sides by sinΘ </li></ul><ul><li>Divide both sides by 1/sinΘ </li></ul>=
9. 9. Elegant Problem Determine Theta Given The Equation Below: = 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. It is obvious that theta = 90 degrees or Π/2 radians Now let’s take a look at a unit circle to see this.
10. 10. Elegant Problem
11. 11. Elegant Problem Now the question did not indicate an interval for theta, so let’s just assume that it wants every angle for when sinΘ=1. Degrees Θ = 90° + K360 where K is an element of I And Radians Θ = Π/2 + K2Π where K is an element of I
12. 12. Congratulations! You’re Done!