30 o. 60 o 1. INSTRUCTIONAL PLAN Day 3

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1 INSTRUCTIONAL PLAN Day 3 Subject: Trigonometry Topic: Special Right Triangles, Problem Solving and Applications of Right Triangles Target Learners: College Students Objectives: At the end of the lesson, students will be able to recognize special right triangles, explain their importance and demonstrate how to use them to make problem solving easier and faster; find the right triangles in a given problem and demonstrate how to use the properties of right triangles to solve mathematical problems; and create and solve problems involving real-life situations with the use of right triangles. Instructional Activity Description of the Activity. Motivation Use the following triangles as a hook: Say, I can give you the exact length of the third side and the all the trigonometric ratios of these triangles without using a calculator. This is possible because these are special triangles. 2. Objective At the end of the lesson, students will be able to: recognize special right triangles. explain the importance of special triangles and demonstrate how to use special right triangles to make problem solving easier and faster. find the right triangle(s) in a given problem and demonstrate how to use the properties of right triangles to solve mathematical problems. create and solve problems involving real-life situations with the use of right triangles. 3. Prerequisite Students will need prior knowledge about angles, triangles, ratio and proportion, exponents and radicals, solutions to algebraic equations, properties of right triangles, Pythagorean Theorem, the six trigonometric ratios, and inverse of sine, cosine and tangent. 4. Information and examples. Show the properties of an isosceles right triangle (45 o X45 o X90 o ). leg:leg:hypotenuse ratio is ; ; ; ; 2. Show the properties of a 30 o X60 o X90 o triangle. leg:leg:hypotenuse ratio is ; ; ; ; 2 30 o ; ; ; ; 60 o

2 3. By showing the students the sine and cosine of complementary angles above, let them arrive at a conclusion that co-ratios are the same for complementary angles. 4. Sample problem: (a) What is the area of an equilateral triangle with side 2 inches? Solution: Draw a line from one vertex perpendicular to the side opposite to that vertex. It will form two 30 o X60 o X90 o triangles. By the property of 30 o X60 o X90 o triangles, we know that the line drawn, which is also the height of the equilateral triangle, will have a length of inches. Thus, the area of the equilateral triangle is inches. (b) A point P lies on a line tangent to a circle of radius 4. If the distance of P to the intersection of the line and the circle is also 4, what is the distance of P to the center of the circle? Solution: Since a tangent line of a circle is always perpendicular to a line from the center of the circle, we will have an isosceles right triangle. Using the properties of isosceles right triangles, we can conclude that the distance from P to the center is. 5. Introduce the concept of Pythagorean Triples and provide the most commonly encountered Pythagorean Triples Build the students vocabulary with the following terms: Special Right Triangles right triangles with measurements and ratios that are easy to remember, and thus reduces the necessity of using calculators or having long solutions. Pythagorean Triples three positive integers that satisfy the Pythagorean Theorem, that is, a 2 + b 2 = c 2 ; a, b, c Є N. Angle of Elevation the angle from the horizontal going up. Angle of Depression the angle from the horizontal going down. 7. Sample problems: (a) Suppose that angle θ is a central angle of a circle of radius (see the figure), show that: i) ii) iii) If θ = 60 o, find AC. ( ) (Source: Sullivan & Sullivan, 2009)

3 (b) A helicopter is hovering over the desert when it develops mechanical problems and is forced to land. After landing, the pilot radios his position to a pair of radar stations located 25 miles apart along a straight road running north and south. The bearing of the helicopter from one station is N 3 E, and from the other it is S 9 E. After doing a few trigonometric calculations, one of the stations instructs the pilot to walk due west for 3.5 miles to reach the road. Is this information correct? (Yes) (Source: McKeague & Turner, 2008) (c) An ecologist wishes to find the height of a redwood tree that is on the other side of a creek, as shown in Figure 9. From point A he finds that the angle of elevation to the top of the tree is 0.7. He then walks 24.8 feet at a right angle from point A to point B. There he finds that the angle between AB and a line extending from B to the tree is What is the height of the tree? (78.9 ft) Technological requirement: Hardware scientific calculator 5. Practice and feedback. Group the class into pairs. Each pair will create 2 real-life situation problems that would involve the use of right triangles to solve. They will answer both problems on a separate sheet of paper. They will then exchange questions with another pair and answer those questions on another piece of paper. Afterwards, the two pairs will confer and share each of their solutions to the problems. Use Worksheet 2: Applications of Right Triangles for this activity. 6. Additional examples Special Angles and their Trigonometric Ratios Source: Sullivan & Sullivan (2009)

4 7. Additional practice and feedback Challenge Questions:. Find r. Answer:.2 in Source: Barnett, R. et al (20) 2. Cost Analysis. A cable television company wishes to run a cable from a city to a resort island 3 miles offshore. The cable is to go along the shore, then to the island underwater, as indicated in the accompanying figure. The cost of running the cable along the shore is P600,000 per mile and underwater, P,000,000 per mile. (a) State the Cost function in terms of θ, where C(θ) is in millions of pesos. (b) Use a graphing calculator to find out what the minimum cost is and what the angle θ is at minimum cost. (P4.4M, 37 o ) Answer: 4.25 ft from the upper-left corner Source: Sullivan & Sullivan (2009) Technological requirement: Hardware scientific calculator, graphing calculator 8. Summary. An isosceles right triangle (45 o X45 o X90 o ) has a leg:leg:hypotenuse ratio of. While a 30 o X60 o X90 o triangle has a leg:leg:hypotenuse ratio is. Their trigonometric ratios are easy to derive and memorize. 2. Co-ratios of complementary angles are equal. 3. Pythagorean triples are positive integers that satisfy the Pythagorean Theorem. If we know two of the integers, it would be easy to figure out the third, as well as all the trigonometric ratios. 4. Applications of right triangles are vast and go beyond trigonometry and geometry. And there is more than one way to solve these problems using right triangles.

5 9. Homework. Review for Unit Test. 2. Journal/blog entry in their CMS pages about what they have learned from the session and their insights on how this new knowledge can be applied in their lives. Evaluation/Assessment: Worksheet and Class participation, both are to be assessed using the Classwork/Participation Rubric (generated from irubric References: Aufmann, Richard N., Barker, Vernon C. and Nation, Richard D. (20). College Algebra and Trigonometry, th ed. Barnett, Raymond A. et al (2008). College Algebra with Trigonometry, 9 th ed. Larson, Ron (202). Algebra and Trigonometry: Real Mathematics, Real People, 6 th ed. McKeague, Charles P. and Turner, Mark D. (2008). Trigonometry, 7th ed. Sullivan, Michael and Sullivan, Michael III (2009). Algebra & Trigonometry, 6 th ed. Prepared by: Cyrus B. Alvarez