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Applications and Trig Identities

• We retain our theme that Algebra is Transformations:  Turning difficult-looking problems into ones we have an idea of how to solve.  In the case of solving right triangles, we can use our knowledge of 2 of the 4 things you need to know to survive Trig:
• Pythagorean Theorem, 
• SOH-CAH-TOA - the definitions of the 3 basic trig functions.
  (we can sometimes also use the Special Right Triangles to check our answers for reasonableness).

• Before starting, though, we need to remember a few very important points:
• Calculator-ready equations:  We should always solve equations algebraically for the unknown (side or angle) before using the calculator for the final step.  Otherwise, we increase the chances of entering the wrong instructions into the calculator.
• Precision:  whenever possible, we should use the original data we were given in any calculations.  Calculating using previously calculated and rounded numbers increases the error we introduce into each step and into the final answer.
• Mode:  we must check/set the mode our calculator is set for (degrees, radians, gradiants, etc.), since the calculator will only do what we've told it to do.

Practice Solving Right Triangles

• This GeoGebra worksheet allows us to move the vertices of right triangles to create problems with sets of known side and/or angles. The check boxes allow us to check our answers and to get hints about the equations we should be using for the different sets of given data.  

• The Elevation/Depression worksheet reminds us that the horizon is a constant; no matter where we look from and to, we consider the horizon lines from both positions (vertices) to be parallel.  So, the angles of elevation and depression are congruent.

• Remember that it is usually easier to convert angle of depression problems into angle of elevation problems, since they match how we usually draw our triangles.
• With word problems, we must be careful that we draw/label our angles and sides in the correct locations.
• All of the advice above for solving right triangles applies to these problems as well.

•  Course and Bearing problems have several differences from the other applications we have been working.  So, they require a great deal of experimentation and practice.  While the basics of right triangle trig, plus a little bit of geometry, are still all we need to work them, the list below highlights some of the differences we need to keep in mind.

  • Angles are measured clockswise from North,
  • The problems are worked in 4 parts:
  • Finding the bearing to an object directly off to our left or right,
  • Determining how far we travel (speed x time),
  • Using the Angle Addition Postulate to find a specific angle back to the object,
  • Using the angle and distance, above, to solve the resulting right triangle.
  • Angles are all 0º and 360º, requiring careful work when crossing through North.

• The Course and Bearing page discusses how we solve them; the follow-on links allow us to experiment and practice.

•  In class, we began with the unit circle and the Pythagorean Theorem to find the basic Pythagorean Identity:

  • cos²ø + sin²ø = 1
    
  • From that we can:
  • solve for cos²ø or sin²ø
  • divide through by cos²ø to produce 1 + tan²ø = sec²ø
  • divide through by sin²ø to produce cot²ø + 1 = csc²ø

• The Identities construction provides us a Geometric picture of how and why these work.

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