Using trig to find the angles, side lengths, and area of triangles

1. Trigonometry and Triangles Michael Schmidt

2. Trigonometry Sine Cosine Tangent Length of triangle legs Angle of triangle corners Area of triangles What we are doing

3. Branch in Mathematics Uses trig functions Triangles Mostly right triangles Uses relationships to find unknowns Trigonometry

4. θ (Theta) Adjacent leg (A) Opposite leg (O) Hypotenuse (H) Key Terms H O θ A

5. SOH: sin θ =𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 CAH: cos θ =𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 TOA: tan θ =𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 SOH CAH TOA

6. sin 37° = cos37° = tan 37° = SOH CAH TOA continued 21𝑓𝑡35𝑓𝑡 28𝑓𝑡35𝑓𝑡 21𝑓𝑡28𝑓𝑡 35ft 21ft 37° 28ft

7. What is given? Which trig function? Finding the side length 12ft sin 30° = 𝑋12𝑓𝑡 X sin 30°1 = 𝑋12𝑓𝑡 X= 6ft 30°

8. Using trig, find unknown tan 20° = 𝑋6𝑚 1 X X= 2.18m 20° 6m X cos 45° = 8𝑖𝑛𝑥 1 45° X= 11.31in 8in

9. What is known? tan θ = 34 Use 𝑡𝑎𝑛−1 𝑡𝑎𝑛−1( 34 ) = θ θ = 36.86° Using trig to find θ 3’ θ 4’

10. sin θ = 10𝑚15𝑚 𝑠𝑖𝑛−1(10𝑚15𝑚 ) = θ θ = 41.81° Solve for θ θ 15m 10m cos θ = 7𝑐𝑚9𝑐𝑚 𝑐𝑜𝑠−1(7𝑐𝑚9𝑐𝑚 ) = θ θ = 38.94° 9cm θ 7cm

11. Area of triangle A = 𝐵𝐻2 A = 15∗102 = 75𝑚2 Finding the Area 10m 15m

12. What is given? What is needed? How is it found? tan60 = 𝐵10 A =17.32𝑐𝑚∗10𝑐𝑚2 Finding area with trig 60° 10cm B =17.32cm =86.6𝑐𝑚2

13. Given Needed sin50 = 𝐻11 A= 13∗8.432 Non right triangles H= 8.43in 11in A=54.8𝑖𝑛2 50° 13in

14. cos 60 = 𝐻22 Pythagorean theorem for the base A= 19.05∗112 Find area of triangle H=11in 60° 22in A=104.78𝑖𝑛2

15. Given Needed B= X+Y tan 45 = 𝑥16 tan 30 = 𝑦16 B=25.24cm Find area of triangle continued Height = 16cm 16cm 45° 30° X Y X=16cm Y=9.24cm A=16∗25.242 =201.92𝑐𝑚2

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 60° how long is the shadow? Story Problems tan 60 = 𝐿6 L=10.39ft 60° 6ft L

17. Story problems 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? sin 52 = 𝐻40 H=31.52ft 40ft No, the ladder will not reach the window. 52°

18. SOH CAH TOA is key Find the Given and Needed Make own right triangle Draw a picture Wrap up