Midterm Analysis: Inverse Trigonometry

Another area of struggle on my PreCalc midterm is inverse trig functions:

sin tan cos

Students had a completed unit circle (radians, degrees and coordinates – they earned them on the exam by completing one on their own the previous week) but could not use a calculator for this section. What types of mistakes do you think students would make?

Feel free to check out the exact mistakes my students made at the Math Mistakes blog.

What I noticed:

While inverse trig was a struggle all year, I was concerned how many students over-thought $sin^{-1}(sin(\dfrac{5\pi}{6}))$ . When my co-worker and I wrote the midterm we thought that question would be a gimme, but student after student evaluated $sin(\dfrac{5\pi}{6})$ (to varying degrees of success) and then found the arcsin. It didn’t occur to me to include many of the correct answers I saw (π/6 and even 17π/6!) on my answer key but luckily I was paying attention when I saw the first one and added solutions to my key as I found them. While 17π/6 isn’t a mistake, I am curious why students didn’t notice that the sine and arcsine cancel.

For the next problem I anticipated students would struggle since inverse tangent isn’t easy to read directly off of the unit circle. Not only do they have to recall that tangent is $\dfrac{sine}{cosine}$ , but they also needed to be able to simplify a fraction within a fraction. The red flag was students who didn’t recognize that there was an intermediate step of simplification; instead they expected there to be an angle where sine=1 and cosine= $\sqrt{3}$ (apparently I didn’t get a picture of that mistake, sorry). The mistake I did photograph was an interesting one though. The student recognized that the -1 meant inverse, however they misattributed it to the inverse of the ratio rather than the inverse of tangent. That shouldn’t be a mistake I see on the midterm though. Especially after explaining countless times that for tangent the input is an angle and the output is a ratio, while for arctan the input is a ratio and the output is an angle.

On the final problem I saw good attempts and students who couldn’t solve it generally forgot that triangle trig was an option. I think those issues were more indicative of a lack of studying than a misconception. Or maybe an inflexibility in their approach? I’ve certainly seen students start a problem only to find that they can’t solve it. At this point they call for help assuming they made a mistake somewhere. Many times their work is accurate and what they actually need is to consider a different method. They could do with a good dose of productive struggle.

Requests for feedback:

I am continually surprised by how much difficulty students have doing anything backwards. In geometry we will study angles formed by parallel lines until they can fill in all 8 angles without a second thought. But then when I give them angles and ask if the lines are parallel, they look at me like I’m asking them if unicorns have purple stripes. The same thing happened here, students are great at finding values of sine, cosine and tangent if I give them the angle. But once I switch it up and ask for the angle instead, they start giving nonsensical answers. What is it about working backwards that students find so challenging? How do you help students recognize the structure?

p.s. Do unicorns have purple stripes?

6 thoughts on “Midterm Analysis: Inverse Trigonometry”

Gregory Taylor (@mathtans) says:

February 11, 2013 at 10:27 pm

I think some of the issue is making connections… though a lot of it is also the notation and terminology used in mathematics. On the first, about #2 you say “they expected there to be an angle where sine=1 and cosine=root(3)” – well, there IS one like that! It’s simply not on the unit circle. So we determine hypotenuse, and then scale the 2 back to 1 using a similar triangle, recheck the sides and there you go. But as you point out, they “generally forgot that triangle trig was an option”, so likely wouldn’t have considered similar triangles at all.

With regards terminology… “The student recognized that the -1 meant inverse, however they misattributed it to the inverse of the ratio” — where what we really mean there is MULTIPLICATIVE inverse, or in other words, what we call the reciprocal. (So did they recognize -1 meant inverse, or did they think it meant reciprocal?) The difference between sine, cosecant, and arcsine is often anything but clear, and the notation sin^-1 doesn’t help. (In my opinion, anyway.)

To try and clarify the point, I actually use my personified mathematics characters. The inverses are the male functions. Does it help? I can’t say for sure, but I don’t think it hinders? (For those who don’t know what I mean, which is everyone, I’m referring to this: https://sites.google.com/site/taylorspolynomials/characters#trigs ) Which may not be for everyone (or anyone), so as an alternative, show the “Use an Inverse, Solve Me Maybe” song, which is here: http://www.youtube.com/watch?v=OsEd7X5XuCU

Also, yes, some unicorns have purple stripes. In particular, Twilight Sparkle from My Little Pony. http://mlp.wikia.com/wiki/Twilight_Sparkle …I don’t watch it, but I suspect the mention may kill what little credibility I had left? I seem to be having a Monday.

Riley John Gibbs says:

February 11, 2013 at 11:26 pm

Logically speaking, you can say almost anything you want about unicorns, since they don’t exist. It’s like saying, “If one plus one equals three, then the sky is maroon.”

I love vacuously true statements.

nik_d_maths says:

February 12, 2013 at 2:21 am

Quick thought on the -1 meaning inverse. I really dislike that notation because of this confusion. I encountered something similar right after we introduced the idea of having the power of a trig function ‘inside’ it; suddenly students thought arcsin was 1/sin and so on. Not sure how to solve that one, but you are not alone!

Joshua Zucker says:

February 28, 2013 at 12:42 pm

Wait, I’m puzzled. You must have a different definition of inverse sine than I do. It sounded like you had 5pi/6 in mind as your answer, but I think inverse sine is a function and that (by somewhat arbitrary but widely agreed convention) it only produces answers in the range -pi/2 to pi/2. What you’re saying here sounds like “square root and squaring cancel”, which is similarly (by somewhat arbitrary but widely agreed convention) only true in a certain range. sqrt(x^2) is not equal to x when x is negative!

I agree that the more important issue here is getting them to be resourceful in using different approaches (unit circle, drawing a triangle, straight computation, using identities, and so on).

Tina C. says:

March 2, 2013 at 6:55 pm

You, sir, make an excellent point! That restriction is not something I emphasized this year. Although, when we solve polynomials I want students to give both the positive and negative values of the square root. Now you’ve got me thinking, this may result in a new post…

Reply

Joshua Zucker says:

March 3, 2013 at 11:43 pm

Ack! I don’t want them taking two values of the square root when they solve polynomials, either. They should go from x^2 = 4 to |x| = 2, would be my preference, or straight to x = +/- sqrt(4) maybe? sqrt(x^2) = |x| is what I try to remind them of in those steps.

Gregory Taylor (@mathtans) says:

February 11, 2013 at 10:27 pm

I think some of the issue is making connections… though a lot of it is also the notation and terminology used in mathematics. On the first, about #2 you say “they expected there to be an angle where sine=1 and cosine=root(3)” – well, there IS one like that! It’s simply not on the unit circle. So we determine hypotenuse, and then scale the 2 back to 1 using a similar triangle, recheck the sides and there you go. But as you point out, they “generally forgot that triangle trig was an option”, so likely wouldn’t have considered similar triangles at all.

With regards terminology… “The student recognized that the -1 meant inverse, however they misattributed it to the inverse of the ratio” — where what we really mean there is MULTIPLICATIVE inverse, or in other words, what we call the reciprocal. (So did they recognize -1 meant inverse, or did they think it meant reciprocal?) The difference between sine, cosecant, and arcsine is often anything but clear, and the notation sin^-1 doesn’t help. (In my opinion, anyway.)

To try and clarify the point, I actually use my personified mathematics characters. The inverses are the male functions. Does it help? I can’t say for sure, but I don’t think it hinders? (For those who don’t know what I mean, which is everyone, I’m referring to this: https://sites.google.com/site/taylorspolynomials/characters#trigs ) Which may not be for everyone (or anyone), so as an alternative, show the “Use an Inverse, Solve Me Maybe” song, which is here: http://www.youtube.com/watch?v=OsEd7X5XuCU

Also, yes, some unicorns have purple stripes. In particular, Twilight Sparkle from My Little Pony. http://mlp.wikia.com/wiki/Twilight_Sparkle …I don’t watch it, but I suspect the mention may kill what little credibility I had left? I seem to be having a Monday.

Riley John Gibbs says:

February 11, 2013 at 11:26 pm

Logically speaking, you can say almost anything you want about unicorns, since they don’t exist. It’s like saying, “If one plus one equals three, then the sky is maroon.”

I love vacuously true statements.

nik_d_maths says:

February 12, 2013 at 2:21 am

Quick thought on the -1 meaning inverse. I really dislike that notation because of this confusion. I encountered something similar right after we introduced the idea of having the power of a trig function ‘inside’ it; suddenly students thought arcsin was 1/sin and so on. Not sure how to solve that one, but you are not alone!

Joshua Zucker says:

February 28, 2013 at 12:42 pm

Wait, I’m puzzled. You must have a different definition of inverse sine than I do. It sounded like you had 5pi/6 in mind as your answer, but I think inverse sine is a function and that (by somewhat arbitrary but widely agreed convention) it only produces answers in the range -pi/2 to pi/2. What you’re saying here sounds like “square root and squaring cancel”, which is similarly (by somewhat arbitrary but widely agreed convention) only true in a certain range. sqrt(x^2) is not equal to x when x is negative!

I agree that the more important issue here is getting them to be resourceful in using different approaches (unit circle, drawing a triangle, straight computation, using identities, and so on).

- Tina C. says:
  
  March 2, 2013 at 6:55 pm
  
  You, sir, make an excellent point! That restriction is not something I emphasized this year. Although, when we solve polynomials I want students to give both the positive and negative values of the square root. Now you’ve got me thinking, this may result in a new post…
  
Joshua Zucker says:

March 3, 2013 at 11:43 pm

Ack! I don’t want them taking two values of the square root when they solve polynomials, either. They should go from x^2 = 4 to |x| = 2, would be my preference, or straight to x = +/- sqrt(4) maybe? sqrt(x^2) = |x| is what I try to remind them of in those steps.

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