lesson 5.3

Homework, Quizzes, and Practice Policy

  • The instructor wants you to have completed all homework that will appear on the quiz before the quiz.
  • Starting next week, quizzes will include material from homework; next week's quiz (5.35) will likely ask you to complete about 75% of it.
  • Roles in class:
    • Tutors/helpers can guide you, but they are not professors.
    • The professor (the instructor) is responsible for teaching you the material.
    • It’s your job to practice at home.
  • Friday nights are used to give you time to catch up in class; layouts are available to assist planning and organization.
  • Memorization vs reference:
    • If you memorize the unit circle, you can brain-dump it onto scratch paper and reference it during tests.
    • The teacher will still expect you to demonstrate understanding, but brain-dumping can help on tests.
  • Unit circle approach example:
    • The teacher uses 30° as a reference angle at the top of the page.
  • Calculator input tips (cosine example):
    • To compute

      ext{cos}(53^ ext{o})

      on a calculator, you can type the angle first (53) and then press the cosine key.
  • Fractions and reciprocal tip:
    • When you have a fraction like rac{1}{ ext{sin}(68^ ext{o})}, typing it as “1 divided by sin(68°)” may not work directly on some calculators.
    • Use the reciprocal function: there is a key for the reciprocal underneath the sine key labeled rac{1}{x}; press that to take the reciprocal.
  • Degree mode vs radian mode:
    • In degree mode, the calculator may misprocess certain inputs if you expect radian results.
    • You need to switch to the correct mode for the problem.
  • How to switch modes:
    • Use the DRG button (to the right of the second function key) to cycle between Degrees (DEG), Radians (RAD), and Grads (GRAD).
    • The screen will show the current mode (e.g., top of screen shows ext{rad} when in radian mode).
  • Composite functions:
    • The homework includes questions on composite functions; the transcript doesn’t provide a detailed definition, but it’s highlighted as a topic you should review.
  • Practical problem context (setup teased):
    • A word problem is used to illustrate trigonometry in a real-world context (e.g., a river width problem and angle-of-view problems).
  • Visual approach and pacing:
    • The instructor sometimes draws two triangles to compare a smaller angle and a larger angle to aid understanding.
    • The idea is to reinforce the idea that different angles require different trigonometric functions for the same kind of measurement (opposite, adjacent, hypotenuse).

Quick reference: general calculation patterns mentioned

  • Tangent-based river width problem:
    • Given an angle and an adjacent distance, find the opposite side (the width of the river):
      an( heta) = rac{ ext{opposite}}{ ext{adjacent}} \ ext{opposite} = ext{adjacent} imes an( heta)
    • Example provided: angle heta = 30^ ext{o}, adjacent = 300 ext{ m}.
    • Calculation:
      b = 300 an(30^ ext{o}) \, ext{m} \ b \, ext{approximately} \, 173.205 \text{ m} \Rightarrow \text{rounded to nearest meter: } 173 \text{ m}.
  • Inputting cosine with a specific angle (53°) example:
    • Enter the angle first, then press the cosine key to evaluate ext{cos}(53^ ext{o}).
  • Reciprocal (1/x) vs direct division:
    • For a fraction like rac{1}{ ext{sin}(68^ ext{o})}, use the reciprocal key (1/x) located under the sine key instead of typing as a direct division in some calculators.
    • This yields the same result as ext{csc}(68^ ext{o}) but via the calculator’s reciprocal function.
  • Mode considerations:
    • Always verify you are in the correct mode for your calculation: degrees vs radians vs grads.
    • Switching modes affects how angles are interpreted and can lead to incorrect results if not aligned with the problem's units.
  • Conceptual reminder on composite functions (homework focus):
    • The topics include composite functions; be prepared to apply the definition and properties in problems (the transcript notes the topic but doesn’t provide the explicit formula).

Practical problem walkthroughs mentioned in the transcript

  • River width problem (specific numbers given):

    • Setup: angle heta = 30^ ext{o}, adjacent length = 300 ext{ m}.
    • Target: find the width of the river (opposite side) labelled as b.
    • Computation:
      an(30^ ext{o}) = rac{b}{300} \ b = 300 an(30^ ext{o}) \ b \approx 173.205 \text{ m}
    • Rounding: the width is approximately 173\ \text{m} (nearest meter).

  • Light-beam to cloud measurement (angle-of-elevation example):

    • Context: measure the angle from the ground to a cloud, with the viewing angle stated as heta = 75^ ext{o}.
    • Trigonometric relationship used: tangent relates opposite and adjacent in a right triangle:
      an( heta) = rac{ ext{opposite}}{ ext{adjacent}} \ ext{opposite} = ext{adjacent} imes an( heta)
    • Physical interpretation in the transcript:
    • The angle is formed from the horizontal ground (the adjacent side) up to the line of sight toward the cloud (the opposite height component).
    • The speaker suggests using two triangles to illustrate how the same problem behaves under a smaller angle vs a larger angle, aiding understanding of solving for both scenarios.
    • In solving for the elusive “height” (opposite), you would typically use tangent when the adjacent distance is known; solving for the hypotenuse would require sine or cosine, but that would not directly answer the stated question.
    • Summary steps for this kind of problem (as described):
    • Identify the angle (theta) and which side is opposite vs adjacent for that angle.
    • Use an( heta) = rac{ ext{opposite}}{ ext{adjacent}} when you want the height (opposite) given a ground distance (adjacent).
    • If you instead needed the line-of-sight distance (hypotenuse), you would use the relationships with sine or cosine, e.g., ext{hypotenuse} = rac{ ext{opposite}}{ ext{sin}( heta)} = rac{ ext{adjacent}}{ ext{cos}( heta)} , but that would be solving a different question.

Study tips and real-world relevance highlighted in the transcript

  • Practice schedule and assessment alignment:
    • Expect quizzes to test material from completed homework.
    • Use Friday night sessions to catch up and prepare using provided layouts.
  • Memorization technique for unit circle:
    • If you memorize, you can quickly write it from memory on scratch paper during tests, which can serve as a reference for the entire test.
  • Calculator literacy:
    • Learn to input cos(53°) by entering 53 then pressing cos.
    • Recognize when to use the 1/x reciprocal key to avoid mis-entering fractions like rac{1}{ ext{sin}(68^ ext{o})} on certain calculators.
  • MODE awareness:
    • Always switch DRG to the appropriate mode (DEG, RAD, GRAD) for your problem to ensure correct angle interpretation.
  • Concept reinforcement through visuals:
    • Drawing two triangles for different angles helps compare how opposite and adjacent sides scale with angle changes.
  • Key takeaway about problem-solving strategy:
    • When you need a length (like the river width), use tangent with the known adjacent length.
    • When you need a distance (like hypotenuse), consider sine or cosine, but ensure you answer the quantity the question asks for.

Summary

  • There is a clear emphasis on timely practice and test preparedness, not just passive listening.
  • The instructor demonstrates concrete calculator habits (input order, reciprocal use, mode switching).
  • Trigonometry problems in the transcript hinge on correctly identifying opposite, adjacent, and hypotenuse relative to the given angle and then applying the appropriate trigonometric function.
  • Real-world examples (river width, angle of elevation to a cloud) illustrate why these concepts matter and how to apply them in practical situations.