Geometry 19 April 20131) 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.

ObjectiveStudents will explore

sine, cosine and tangent ratios.Students will take notes, do and investigation.

ACT Testing on April 23- ONLY JUNIORSAll other students will have the day off school

HOMEWORK DUE TUESDAY midnightHOMEWORK DUE TUESDAY midnightSpring Project ProposalSpring Project Proposal

DUE Wed/ Thursday“Why did the mural painter…”

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?

trignometry

“trigon” means triangle“metry” means “measurement”

Trigonometry is a math field Trigonometry is a math field based on side ratios based on side ratios

in RIGHT TRIANGLES!!!in RIGHT TRIANGLES!!!

http://en.wikipedia.org/wiki/Uses_of_trigonometry

Right Triangle Vocabularyhypotenuse- the longest sideadjacent 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

ANGLEANGLE

Adjacent

Opposite

How do you find O,A and H?

1. first find hypotenuse- the longest side and the side across from the right angle

hypotenuse

Finding the adjacent side2. 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 sideHINT- put one finger on the angle and one finger on the right

angle. The side between them is “ADJACENT”.

hypotenuse

Finding the adjacent side

2. angle

adjacent sidehyp

otenuse

Finding the opposite side

3. the remaining side is the OPPOSITE side(side opposite the angle)

angle opposite side

hypotenuse

adjacent

Finding the opposite side

3.

angle

opposite side

hypotenuse

adja

cent

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 minuteTell your partner what you wroteCan you add to your notes?

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 2nd angle’s adjacent.

PRACTICE

Complete handout correctly identifying each side of the RIGHT TRIANGLE as H- hypotenuseA- adjacent to marked angleO- opposite marked angle

INVESTIGATION- TRIG RATIOSCalculating ratios for similar triangles

NOTENOTE1) all the triangles on each page are right triangles2) all the marked angles on a single page are the same size– verify this with patty paper3) therefore, all the triangles on each page are ________________ and all the side lengths are____________________by the ___ Similarity Conjecture

Activity-Calculating ratios for similar trianglesEXPECTATIONS- 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.

investigation resultsGROUPS present their table- random call1. 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

the trig functions sine, cosine and tangent are RATIOS

of side lengths ofright triangles

Since right triangles with the same acute with the same acute angle measureangle measure are similarsimilar (AA Similarity Conjecture)

these ratios are PROPORTIONALthese ratios are PROPORTIONALand their decimals are equivalent,and their decimals are equivalent,

NO MATTER THE SIDE LENGTHSNO MATTER THE SIDE LENGTHS

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