Trigonometric Identities

I uploaded some Trigonometric Identities Notes with important formulaes: https://www.dropbox.com/s/u49108mat5k2kb1/Important%20Trigonometric%20Identities%20NOtes.pdf







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Darryl Lam

3 Rules necessary for an Ambiguous Sine Rule Case

Ambiguous case of the sine rule is when there are two possible triangles that satisfies the given information.

For example,

sin R / r = sin Q / q
Put in the values we know:sin R / 41 = sin(39º)/28
Multiply both sides by 41:sin R = (sin39º/28) × 41
Calculate:sin R = 0.9215...
Inverse Sine:R = sin-1(0.9215...)
R = 67.1º




However, sin (112.9) = sin 67.1. As such, you can have the two triangles shown below.



The three conditions required for the ambiguous case to happen (before this we need to have two sides e.g. b & a and the non-included angle)

 a < c  
 a < perpendicular ht. of point C to c
given angle (the non-included one) is acute







QUIZ B (SHORT ANSWER)

QUIZ B: to be done on foolscap papers and submit

QUESTION 1



QUESTION 2



QUESTION 3



QUESTION 4

QUIZ A (MCQ)

SUMMARY - TRIGONOMETRY EM

SUMMARY OF TRIGONOMETRY 
(ELEMENTARY MATHEMATICS)



Trigonometry (Bearing - Application) (GROUP TASK 3) (hint)

Activity 1

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Activity 2

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Activity 3

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Activity 4




Trigonometry (Bearing - Application) (GROUP TASK 3)


ACTIVITY 3

Objective:
To provide students to apply the concept of Trigonometry Bearing using a real world context.
To enable students to apply the procedural skills as well as the Mathematics Problem solving Heuristics.
To create a Collaborative working scenario in an authentic learning environment

Task:
Each group will be assigned a Real World problem.
Your product should have the following:
A) Assumptions made
B) Visual representation of problem
C) Solution (showing explicitly the Heuristics used)
D) Analysis of suitability of solution (why do you think the solution is feasible/logical ?)

Posting:
Submit your e-solution in the Group Page and Label it as 
TRIGONOMETRY - BEARING

GROUP'S PROBLEM SCENARIOS
=================================================================
GROUP 1

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GROUP 2

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GROUP 3

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GROUP 4




Trigonometry (Bearing - Application) (GROUP TASK 1)

ACTIVITY 2

Objective:
To enable students to apply the procedural skills as well as the Mathematics Problem solving Heuristics.

Task:
Study the problem given and analyse the solution attached.
Use the Mathematical Heuristics techniques done in previous lessons as a guide.
Post as comment:
  What do you think is the heuristics used in solving the problem? Justify your answer




Bearings (Introduction)

Bearing (introduction)

ACTIVITY 1
Objective
To have a better appreciation of the application of trigonometry in the real world by using a collaborative tool (Linoit).

Task 1
Individually review the resources included in the linoit and answer the questions included. Post your findings/responses/resources in the Linoit.





SINE Rule (special cases)

Follow up from the class activities
source: http://www.mathsisfun.com/algebra/trig-sine-law.html









Sine Rule ~ Esther, Dawn and Luke

 Example :
 


Trigonometry Activity 2: SINE and COSINE Rules

The following videos show the derivation of (i) SINE rule and (ii) COSINE rule.
The visual representations are not supported by auditory explanation.

Task
Your task, as a group, is to incorporate the explanation from Mathematical perspective using either Viva Voce technique or clear and concise text explanation.

SINE RULE





COSINE RULE


Trigonometry Activity 2

Using the same technique of using Mathematical heuristics, solve the following problem and post the (i) assumptions (if any) (ii) process and (iii) solution in the Group's page.



The diagram shows a loading ramp used for ferries. It is supported by cables that are fed out from a tower and fixed half way along the ramp A.
The ramp must be fixed so that it is horizontal at high tide.
The maximum angle of depression allowed, at low tide on the ramp is 18 degrees.

(a) Explain, using simple mathematical terms, the above problem.
(b) At which of these ports could this ramp be used?
     Justify your answer fully from Mathematics and engineering points of view?

     Harbour            Tidal range
     Arden               4.28 m
     Jameston           2.86 m
     Daleen              3.85 m
     Palter                3.54 m

(c) A sensor is to be fitted to the cable sounding alarm when the slope reaches 18
     degrees.
     How far from A should the sensor be fixed?
     Explain the reason(s) for having this alarm.

Heuristic Problem solving in Mathematics (Activity 1)



Problem
Two men facing a tall building notice the angle of elevation to the top of the building to be 30o and 60o respectively. If the height of the building is known to be 
h =120 m, find the distance (in meters) between the two men.


Heuristic Problem Solving
Stage 1
Question any assumptions
Key in your assumption/s in the linoit individually.




Questions surfaced from Stage 1:

Stage 2
Solve the Problem Individually (my perspective)
Once your assumptions and hypothesis have been clarified, start working on the problem individually. No discussion with anyone for this stage.
Added Scenario:
You are the Civil and Structural Engineer and the 2 men are the land surveyors from a Company. You have to pit your skills and knowledge with engineers from other companies. 

Stage 3
Collaboration - seeking multiple perspectives
In the big group of 5-6 students, present and discuss your solution with the group. 
Focus on the assumptions/hypothesis, Process and the validity of the final solution(s).
Post your Group's solution as a New Post with the following criteria:
Post Title: Heuristic Solution (Group number)
Present the following:
- Assumptions Made
- Process (how you solved the problem) - what method/approach you used eg diagrams, on-line resources
- Final Solution

Stage 4
Peer Assessment
Review the solution(s) from other groups. 
To facilitate process, follow the following:
Group 1 to comment on Group 2
Group 2 to comment on Group 3
Group 3 to comment on Group 4
Group 4 to comment on Group 1

Stage 5
Self Assessment
Review the comments made by the other group.
Be prepared to defend or justify your POV.
However, if an error has been committed - learn from it.

So ... what is there a FINAL solution that all the engineers could agree upon? Justify your answer.