This week I finished the unit Right Triangles and Trigonometry. There were a number of videos/exercises I had to get through to finish the unit, however, looking back, I’m a bit disappointed with my progress and think I could have gotten more done. To be fair though, I most likely spent >6.5 hours on KA which is more than my goal of spending 5 hours per week working through KA AND writing the blog. I think I currently average about 7 hours per week on KA and 1.5 hours writing each blog so I suppose I shouldn’t be too disappointed in myself…

I began the week working through basic SOH-CAH-TOA trig problems. These problems would show me a picture of a right triangle and give me one of the non-90 degree angles and the length of one of the sides and ask me to solve for the length of another side. These questions helped me get a feel for the cross multiplication involved and literally how to enter numbers such as sin(72 degrees) into my calculator. I don’t have a scientific calculator and have been using my iPhones calculator which works fine but I find can be a bit confusing when entering Sine, Cosine, and Tangent numbers/angles into it.

I then learned about Inverse Trigonometry Functions. These questions still involved using SOH-CAH-TOA but, instead of giving me an angle and the length of a side, I was given the length of two sides and asked to find one of the non-90 degree angles. This is done by taking the same ratios used in SOH-CAH-TOA and multiplying them by the inverse trig function. These are written as:

• sin^-1(opp/hyp) = “Theta” (Again, I cannot write the symbol for “Theta” – a circle with a horizontal line through it – which seems to be the symbol most often of used to denote an unknown angle)
  • sin^-1 = “arcsin”
• cos^-1(adj/hyp) = “Theta”
  • cos^-1 = “arccos”
• tan^-1(opp/adj = “Theta”
  • tan^-1 = “arctan”

It’s worth noting that the ^-1 used in inverse trig functions does not indicate a negative exponent of one is being applied to sin, cos or tan. It’s simply the method of denoting the inverse function or all three.

Two new names I learned for angles are “Angle of Elevation” and “Angle of Depression”. As far as I know, they both refer to measures of angles that begin from a horizontal line (i.e. a flat line that has no slope), but I may be wrong on this. An Angle of Elevation moves “upwards” from the horizontal line (i.e. counter clockwise from the horizontal line) and an Angle of Depression moves downwards from the horizontal line (i.e. clockwise from the horizontal line).

I learned that the sin(Theta) = cos(90 degrees – Theta). This is because the Sine lines of a given angle (i.e. sin = opposite/hypotenuse), are the exact same Cosine lines (i.e. cos = adjacent/hypotenuse) and Cosines angle is the “compliment” of Theta (i.e. Cosines angle is 90 degrees – “Theta”). I find this concept tricky to put into works. I still don’t have a great understanding of how it works but my hope is that, after I spend more time working through these questions, I’ll better understand how it works and will be able to explain it more clearly.

The last two things I learned about was the Law of Sines and the Law of Cosines. Up until this point, I had only used trigonometry with right angle triangles. Using the Law of -Sines and -Cosines, I was taught how to “solve” a triangle (i.e. how to find the missing values of certain sides and/or angles) that isn’t a right angle triangle. Here’s how they work:

• Law of Sines
  • States that, “the ratios between the Sine of an angle in a triangle and it’s opposite side is the same ratio between the other two angles their respective opposite sides.”
  • Knowing this, you can apply cross multiplication to the Law of Sines to find out a missing side or angle.
    • sin(a)/A = sin(b)/B
      • sin(82 degrees)/11 = sin(44 degrees)/B
      • B = (11)sin(44 degrees)/sin(82 degrees)
      • B = ~7.7
• Law of Cosines
  • Is used to find the value of an opposite-side when given Side, Angle, Side OR used to find any angle when given all three side lengths.
  • C^2 = A^2 + B^2 – 2(A)(B)cos(“Theta”)
    • Missing side ex.
      • C^2 = 5^2 + 6^2 – 2(5)(6)cos(74 degrees)
      • C^2 = 25 + 36 – (60)cos(74 degrees)
      • C^2 = 61 – ~16.54
      • {C^2} = {~44.46}
      • C = ~6.7
    • Missing angle ex.
      • 9^2 = 7^2 – 13^2 – 2(7)(13)cos(“Theta”)
      • 81 = 49 + 169 – (182)cos(“Theta”)
      • -132 = (-182)cos(“Theta)
      • -132/-182 = (-182)cos(“Theta)/-182
      • ~0.752 = cos(“Theta”)
      • cos^1(~0.752) = “Theta”
      • “Theta” = ~41 degrees

Going through the unit test, I struggled with 30-60-90 triangles and had to retake the test after getting one of those questions wrong. I’m still having a hard time A) remembering the ratios and B) applying them with cross multiplication to find the missing values of sides.

Considering there is an entire course dedicated to trigonometry coming up, I think it’s ok that I don’t fully understand how trigonometry works at this point. My guess is this unit is intended to introduce me to these concepts so I don’t feel caught off guard when I begin the trig course.

This coming week I’m hoping to get through two units, Solid Geometry (0/600 M.P.) and Analytic Geometry (0/900 M.P.) They both look fairly small so I should be able to grind through them. After that, there’s a big unit on circles (0/1700 M.P.) and then the course challenge to complete the course. The goal is to finish the unit by week 26, i.e. the half-year mark.

I can’t think of anything clever to say to finish this blog so… bye.