The document discusses different methods for determining the width of a river without directly measuring it, using only a meterstick and protractor. It presents 6 solutions that use trigonometric concepts like similar triangles, congruent triangles, and angle ratios. It then discusses representing the relationships between quantities as functions and generating additional problem cases. Questions are provided at the end to discuss which solutions students prefer and how other cases could be modeled.

1. Teaching trigonometry through problem solving An introductory lesson on tangent and cotangent Erlina R. Ronda UP NISMED

2. Erlina R Ronda http://keepingmathsimple.wordpress.com 2 PROBLEM How would you determine the width of a river if you cannot measure it directly? You cannot cross the other side, too, but you have with you a meterstick and a big protractor.

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4. 4 Solution 1: using 45-degree angle river x 45o k x = k Erlina R Ronda http://keepingmathsimple.wordpress.com

5. 5 mo mo Solution 2: congruent triangles river x k b x = b Erlina R Ronda http://keepingmathsimple.wordpress.com

6. 6 Solution 3: using 60 or 30-degree angle x river 60o k Erlina R Ronda http://keepingmathsimple.wordpress.com

7. 7 Solution 4: similar triangle, geometric mean river x mo 90o - mo a k Erlina R Ronda http://keepingmathsimple.wordpress.com

8. 8 mo Solution 5: similar triangle river x b k a Erlina R Ronda http://keepingmathsimple.wordpress.com

9. 9 mo mo Solution 6: similar triangle river x k a b Erlina R Ronda http://keepingmathsimple.wordpress.com

10. 10 Questions for discussion 2) What quantities are involved in the solutions? 1) Which solution do you like most? Like least? WHY? 3) Which quantities are related as function? 5) What are other ways of representing this function? 4) How can other cases be generated for this function? Erlina R Ronda http://keepingmathsimple.wordpress.com

11. Through use of grid and protractor, students can get the value of the ratios 11 40o Relationship 2: angle ->adj/opp Relationship 1: angle ->opp/adj Erlina R Ronda http://keepingmathsimple.wordpress.com

12. Erlina R Ronda http://keepingmathsimple.wordpress.com 12 Table of values

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15. 15 Erlina R Ronda http://keepingmathsimple.wordpress.com Going back to the initial problem … How will you use the idea of the ratio of the shorter sides in a right triangle to solve the problem about the width of the river? How does this solution compare with the one using similar triangles?

16. 16 B Given rtΔABC. Let A = xoside opposite A = a;side adjacent to A = b. a xo A C b Let g: x -> This mapping is called the cotangent function. It is defined by the equation cotangent x = g(x) or cotangent x = . Let f: x -> This mapping is called the tangent function. It is defined by the equation tangent x = f(x) or tangent x = . Erlina R Ronda http://keepingmathsimple.wordpress.com