1. Geometry 19 April 2013 1) Trade your multiple choice question with a partner. Work your partner’s question. Give feedback. 2) Work CYLINDER volume and surface area problem on yellow handout. Place your Similarity Summary of Concepts and Multiple Choice Question on top of group folders.

2. Objective Students will explore sine, cosine and tangent ratios. Students will take notes, do and investigation. ACT Testing on April 23- ONLY JUNIORS All other students will have the day off school HOMEWORK DUE TUESDAY midnight Spring Project Proposal DUE Wed/ Thursday “Why did the mural painter…”

3. You are on a boat tour around a tropical island. You see a man at the top of a cliff, diving into the water below. You know that your boat is 500 feet from the base of the cliff (your tour guide tells you) and you estimate that the angle that you look up to the top of the cliff is 30⁰ from horizontal. How high is the cliff?

4. trignometry “trigon” means triangle “metry” means “measurement” Trigonometry is a math field based on side ratios in RIGHT TRIANGLES!!! http://en.wikipedia.org/wiki/Uses_of_trigonometry

5. Right Triangle Vocabulary hypotenuse- the longest side adjacent side- the side that is not the hypotenuse that helps form the angle; adjacent means “next to” opposite side- the side that does not help form the angle; side “across” the triangle Hypotenuse ANGLE Adjacent Opposite

6. How do you find O,A and H? 1. first find hypotenuse- the longest side and the side across from the right angle hypotenuse

7. Finding the adjacent side 2. determine which side helps make the angle for which you determining the trig ratios that is NOT the hypotenuse this is the ADJACENT side angle adjacent side HINT- put one finger on the angle and one finger on the right angle. The side between them is “ADJACENT”. hypotenuse

8. Finding the adjacent side 2. angle adjacent side hypotenuse

9. Finding the opposite side 3. the remaining side is the OPPOSITE side (side opposite the angle) angle opposite side hypotenuse adjacent

10. Finding the opposite side 3. angle opposite side hypotenuse adjacent

11. Stop and Jot - 2 minutes Write down what your just learned about Trig, and identifying hypotenuse, adjacent side and opposite side. Make a sketch and label it. Turn and talk- 1 minute Tell your partner what you wrote Can you add to your notes?

12. Your opposite is my adjacent HYPOTENUSE is always the hypotenuse! Opposite and adjacent DEPEND on the angle. It is a matter of perspective! Your left is my right. One angle’s opposite is the 2 nd angle’s adjacent.

13. PRACTICE Complete handout correctly identifying each side of the RIGHT TRIANGLE as H- hypotenuse A- adjacent to marked angle O- opposite marked angle

14. INVESTIGATION- TRIG RATIOS Calculating ratios for similar triangles NOTE 1) all the triangles on each page are right triangles 2) all the marked angles on a single page are the same size– verify this with patty paper 3) therefore, all the triangles on each page are ________________ and all the side lengths are ____________________ by the ___ Similarity Conjecture

15. Activity-Calculating ratios for similar triangles EXPECTATIONS- GROUP ACTIVITY 1. Each person in your group needs to measure at least two DIFFERENT triangles. 2. Measure each side CAREFULLY to the nearest millimeter. 3. Complete the worksheet writing the ratios as a fraction and decimal. 4. Share your results with your group- each person filling in their worksheet completely. 5. Calculate the mean values for each ratio rounding to 2 decimal places.

16. investigation results GROUPS present their table- random call 1. what do you notice about the ratios when the triangle sizes are different but the acute angles are the same? 2. Trig ratios only work for right triangles. WHY ARE THE RATIOS THE SAME for different triangles with the same angle measures? Explain this using our similarity conjectures. Similar triangles – corresponding angles congruent --corresponding sides proportional

17. with the same acute angle measuresimilar these ratios are PROPORTIONAL and their decimals are equivalent, NO MATTER THE SIDE LENGTHS the trig functions sine, cosine and tangent are RATIOS of side lengths of right triangles Since right triangles with the same acute angle measure are similar (AA Similarity Conjecture) these ratios are PROPORTIONAL and their decimals are equivalent, NO MATTER THE SIDE LENGTHS

18. debrief 1. what do you notice about the ratios when the triangle sizes are different but the acute angles are the same? 2. Trig ratios only work for right triangles. Explain this using our similarity conjectures. Similar triangles – corresponding angles congruent --corresponding sides proportional