Trigonometry and triangles


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Trigonometry and triangles

  1. 1. Trigonometry and Triangles<br />Michael Schmidt<br />
  2. 2. Trigonometry<br />Sine Cosine Tangent<br />Length of triangle legs<br />Angle of triangle corners<br />Area of triangles<br />What we are doing<br />
  3. 3. Branch in Mathematics<br />Uses trig functions<br />Triangles<br />Mostly right triangles<br />Uses relationships to find unknowns <br />Trigonometry<br />
  4. 4. ΞΈ (Theta)<br />Adjacent leg (A)<br />Opposite leg (O)<br />Hypotenuse (H)<br />Key Terms<br />H<br />O<br />ΞΈ<br />A<br />
  5. 5. SOH: sin ΞΈ =π‘‚π‘π‘π‘œπ‘ π‘–π‘‘π‘’π»π‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’<br />CAH: cos ΞΈ =π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘π»π‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’<br />TOA: tan ΞΈ =π‘‚π‘π‘π‘œπ‘ π‘–π‘‘π‘’π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘<br />Β <br />SOH CAH TOA<br />
  6. 6. sin 37Β° =<br />cos37Β° =<br />tan 37Β° =<br />SOH CAH TOA continued<br />21𝑓𝑑35𝑓𝑑<br />28𝑓𝑑35𝑓𝑑<br />21𝑓𝑑28𝑓𝑑<br />Β <br />35ft<br />21ft<br />37Β°<br />28ft<br />
  7. 7. What is given?<br />Which trig function? <br />Finding the side length<br />12ft<br />sin 30Β° = 𝑋12𝑓𝑑<br />Β <br />X<br />sinΒ 30Β°1 = 𝑋12𝑓𝑑<br />Β <br />X= 6ft<br />30Β°<br />
  8. 8. Using trig, find unknown<br />tan 20Β° = 𝑋6π‘š<br />Β <br />1<br />Β <br />X<br />X= 2.18m<br />20Β°<br />6m<br />X<br />cos 45Β° = 8𝑖𝑛π‘₯<br />Β <br />1<br />Β <br />45Β°<br />X= 11.31in<br />8in<br />
  9. 9. What is known?<br /> tan ΞΈ = 34<br />Use π‘‘π‘Žπ‘›βˆ’1<br />π‘‘π‘Žπ‘›βˆ’1( 34 ) = ΞΈ<br />ΞΈ = 36.86Β°<br />Β <br />Using trig to find ΞΈ<br />3’<br />ΞΈ<br />4’<br />
  10. 10. sin ΞΈ = 10π‘š15π‘š<br />π‘ π‘–π‘›βˆ’1(10π‘š15π‘šΒ ) = ΞΈ<br />ΞΈ = 41.81Β°<br />Β <br />Solve for ΞΈ<br />ΞΈ<br />15m<br />10m<br />cos ΞΈ = 7π‘π‘š9π‘π‘š<br />π‘π‘œπ‘ βˆ’1(7π‘π‘š9π‘π‘šΒ ) = ΞΈ<br />ΞΈ = 38.94Β°<br />Β <br />9cm<br />ΞΈ<br />7cm<br />
  11. 11. Area of triangle<br />A = 𝐡𝐻2<br />A = 15βˆ—102 = 75π‘š2<br />Β <br />Finding the Area<br />10m<br />15m<br />
  12. 12. What is given?<br />What is needed?<br />How is it found?<br />tan60 = 𝐡10<br />A =17.32π‘π‘šβˆ—10π‘π‘š2<br />Β <br />Finding area with trig<br />60Β°<br />10cm<br />B =17.32cm<br />=86.6π‘π‘š2<br />Β <br />
  13. 13. Given<br />Needed<br />sin50 = 𝐻11<br />A= 13βˆ—8.432<br />Β <br />Non right triangles<br />H= 8.43in<br />11in<br />A=54.8𝑖𝑛2<br />Β <br />50Β°<br />13in<br />
  14. 14. cos 60 = 𝐻22<br />Pythagorean theorem for the base<br />A= 19.05βˆ—112<br />Β <br />Find area of triangle<br />H=11in<br />60Β°<br />Β <br />22in<br />A=104.78𝑖𝑛2<br />Β <br />
  15. 15. Given<br />Needed<br />B= X+Y<br />tan 45 = π‘₯16<br />tan 30 = 𝑦16<br />B=25.24cm<br />Β <br />Find area of triangle continued<br />Height = 16cm<br />16cm<br />45Β°<br />30Β°<br />X<br />Y<br />X=16cm<br />Y=9.24cm<br />A=16βˆ—25.242<br />Β <br />=201.92π‘π‘š2<br />Β <br />
  16. 16. A 6ft tall man is standing in front of a light. The light is casting a shadow. If the angle of depression at the man’s head is 60Β° how long is the shadow?<br />Story Problems<br />tan 60 = 𝐿6<br />Β <br />L=10.39ft<br />60Β°<br />6ft<br />L<br />
  17. 17. Story problems<br />There is a window 33ft up a building and the only ladder is 40ft long. For safety reasons the ladder is leaned against the building at 52°. Will the ladder reach the window?<br />sin 52 = 𝐻40<br />H=31.52ft<br /> <br />40ft<br />No, the ladder will not reach the window.<br />52°<br />
  18. 18. SOH CAH TOA is key<br />Find the Given and Needed<br />Make own right triangle<br />Draw a picture<br />Wrap up<br />