You are going to solve this as you would an algebraic equation. First, add 20 to each side then divide by two. This leaves you with cosecant of theta equals 2. Cosecant is the reciprocal of sine, so cosecant of theta equals 2 becomes sine of theta equals1/2. On the unit circle, sine is the "y" coordinate, so you want to look for where "y" is a positive 1/2. This happens at π/6 and 5π/6.
Graph the equation
The first step to graphing is to determine the equation's amplitude, period, phase shift, and vertical shift. Always start graphing with five "nice" theta points. These will be the "x" values of the graph. These are 0, π/2, π, 3π/2, 2π. You will apply any transformations to these values. The phase shift is subtracting π/3 from each value because it shifted to the left by a value of π/3. The work for this is shown above. The "y" values for a sine graph are 0, 1, 0, -1, 0. Just like with the theta values, any transformations will be applied to these values.
This is what the graph will look like after being graphed. You can see the phase shift to the left and the vertical shift down.
Solving for inverse functions
Sine is opposite over hypotenuse, so 12 is one leg and 13 is the hypotenuse. Then, use Pythagorean theorem to solve for the other leg, which 5. Tangent is opposite leg over adjacent leg, so the solution is 12/5.
This problem begins the same as the one above. Because it is inverse of 2 tangent, you use the double angle formula. Plug in the sine and cosine values from the triangle and multiply across. The solution is 24/25. How many radians do the minute and hour hand move from noon to 5:30?
Since there are 12 chunks on the clock, each one is 1/12 of the circle. To find the radian measure, multiply 1/12 by 2π. Each chunk is π/6. The hands move 5π/6 radians when it's 5:00. 5:30 is halfway between 5 and 6. To find that radian measure, multiply 1/2 by π/6 to get π/12. To find the total amount of radians the hands travel, add 5π/6 and π/12 to get 11π/12. This means the minute and hour hands travel 11π/12 radians from noon to 5:30.
Reflection:
I chose to do two solving problems because solving was something I really enjoyed during this class. I really enjoy algebra and this was similar to solving in algebra. It was kind of like a puzzle finding the theta value you just solved for.
I did a graphing problem because that was an area I struggled with. Doing that problem and explaining it really helped me understand how to graph more. I needed to know how to do the problem, and how to explain it, which is something I wasn't able to do before.
I did two inverse problems because I also enjoyed doing those. It brought back some triangle trigonometry, which was something I enjoyed from geometry. I also liked how you could use, for example, sine inverse to find tangent.
My last problem was a clock problem. I really struggled with these when we first learned them. After going away from them and coming back, I'm able to understand them easier. I really like how you have to find out what fraction of an hour the minute hand has traveled to find the radian measure.
For my problems, I chose two that I really enjoyed doing and two that I really struggled with. Making up problems for something I had struggled with was a huge challenge for me, but it really helped me to understand the type of problem a lot more. Doing two problems that I understood and enjoyed was more fun for me because I could work backwards to figure out a problem.
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