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# Similar triangles and trigonometric ratios

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### Similar triangles and trigonometric ratios

1. 1. O B J E C T I V E SS T U D E N T S W I L L E X P L O R E T H E C O N N E C T I O N B E T W E E N S I M I L A RR I G H T T R I A N G L E S A N D T R I G O N O M E T R I C R A T I O SS T U D E N T S W I L L F O R M A C O N J E C T U R E A B O U T T H ER E L A T I O N S H I P B E T W E E N S I N E A N D C O S I N E O FC O M P L E M E N T A R Y A N G L E S I N A R I G H T T R I A N G L ES T U D E N T S W I L L A P P L Y S I N E , C O S I N E , A N D T A N G E N T R A T I O S T OS O L V E R I G H T T R I A N G L E SSimilar Triangles and TrigonometricRatios
2. 2. Review of Proving Triangles Similar Three ways to prove triangles similar SSS If all three sides on two triangles are proportional, then thetriangles are similar SAS If two sides are proportional and the included angles arecongruent, then the triangles are similar AA If two angles are congruent, then the two triangles are similar
3. 3. Terms to know for TrigonometryθReference AngleOpposite side(o)Adjacent Side(a)Hypotenuse(h)
4. 4. θReference AngleOpposite side(o)Adjacent Side(a)Hypotenuse(h)Trigonometric RatiosSineSin θ = ohCosineCos θ = ahTangentTan θ = oa
5. 5. Ways to MemorizeTangentTan θ = oaSineSin θ = ohCosineCos θ = ahOhio Has A Heroic Offense AgainSoh Cah Toa
6. 6. Writing the Trigonometric Ratiosx51213(o)(a)(h)
7. 7. Now lets look at the other acute angleθ51213(a)(h)(o)What do you notice about the sine, cosine, and tangentfrom the two acute angles in the same right triangle?x
8. 8. Using Trigonometry RatiosStep 1: Identify which trigonometricratio we are using based on thesides given38°15xOh=sinSin =Step 2: Plug the numbers into the ratio
9. 9. Solvesin 38°=.6157 =.6157 x = 15x= 24.3615x38°x1515xNOTE** Sin 38°=.6157Now, what if we want to find the missing sidePYTHAGOREAN THEOREM!!!
10. 10. Inverse FunctionsWhat if we need to find the angle?Step 1: Identify which trigonometricratio we are using based on thesides givenStep 2: Plug the numbers into the ratioTan =x816Oa=tan
11. 11. Solving Inverse FunctionsTan =To solve, switch the x and the , making the tangentits inverseTan-1 =Now use your Calculators:26.57 °=x816816x
12. 12. ApplicationPythagoras is walking up the stairs. For fun, he decidesto do a little math along the way. Before walking upthe stairs, he measures that the bottom of the stairsmake a 50° angle with the ground. If the stairs reach6.3 meters on the wall, can you tell Pythagoras thelength of the stairs.** Try on your Own
13. 13. Answersin 5o° =.76604 =.76604 x = 6.3x= 8.22 meters50°6.3xx6.3x6.3
14. 14. What did we learn today? The three trigonometric ratios, sine, cosine, andtangent How to use the trigonometric ratios to solve formissing sides and angles Practical applications of their uses
15. 15. Exit Slip Find the missing angles and sides2753°